aa r X i v : . [ m a t h . K T ] M a r THE CONSTRUCTION OF DIRAC OPERATORSON ORIENTIFOLDS
SIMON KITSON ∗ Abstract. Mo tivated by Wigner’s theorem, a canonical construc-tion is described that produces an Atiyah-Singer Dirac operatorwith both unitary and anti-unitary symmetries. This Dirac oper-ator includes the Dirac operator for KR -theory as a special case,filling a long-standing gap in the literature. In order to make theconstruction, orientifold Spin c -structures are defined and classifiedusing semi-equivariant Dixmier-Douady theory, and analogues ofseveral standard theorems on the existence of Spin c -structures areproved. C ontents Orientifold Groups . . . . . . . . . . . . . . . . 42.2 Orientifolds . . . . . . . . . . . . . . . . . . . . 72.3 Orientifold Bundles . . . . . . . . . . . . . . . . 72.4 Operations on Orientifold Bundles . . . . . . . 102.5 Classification of
Spin c -Structures on Orientifolds 12 A.1 Semi-equivariance and Associated Bundles . . 32A.2 Semi-equivariant Connections . . . . . . . . . . 33
References 35
1. Intr oduction
This paper uses new results on semi-equivariant Dixmier-Douady the-ory [21] to determine the orientiation conditions that allow the con- ∗ I would like to thank the Mathematical Sciences Institute of the Australian Na-tional University for the postdoctoral fellowship which supported this research. SIMON KITSON struction of Atiyah-Singer Dirac operators with both linear and anti-linear symmetries. The construction will be described in detail. Theexistence of Dirac operators with linear/anti-linear symmetries is ba-sic to the compatibility between index theory and Wigner’s Theorem.It also fills the gap in the literature regarding the existence of Diracoperators for KR -theory. In addition, it seems likely that such op-erators have important applications in orientifold string theories andcondensed matter physics.In the present context, the term orientifold will refer to a manifoldequipped with an action of a group Γ which, in turn, is equipped witha homomorphism ǫ : Γ → Z . This small amount of extra structureis used to define unitary/anti-unitary actions of Γ on complex vectorbundles over the orientifold. An element γ ∈ Γ acts via a unitary mapif γ ∈ Γ + : (cid:3) k er ( ǫ ) , or an anti-unitary map if γ ∈ Γ − : (cid:3) Γ \ Γ + . Thesevector bundles will be described as orientifold bundles . Note that the setof orientifold bundles over an orientifold depends on the embedding Γ + ֒ → Γ . More generally, the term orientifold will be used as an adjectiveto describe objects carrying, or compatible with, unitary/anti-unitaryactions. For example, the Dirac operator mentioned above acts betweenorientifold bundles in an equivariant manner and will be described asthe orientifold Dirac operator .The construction of the orientifold Dirac operator depends on anunderstanding of the global topology of complex vector bundles withanti-unitary symmetries. The main obstacle to understanding the con-ditions under which an orientifold Dirac operator exists is the failureof equivariant transition cocycles and cohomology to accomodate anti-linear symmetries. This obstacle was overcome in [21] by introduc-ing semi-equivariant transition cocycles, which simultaneously gener-alise Wigner’s corepresentations [29, pp. 334-335] [16, pp. 169-172] andequivariant transition cocycles. Using the results of [21], the obstruc-tion to the existence of an orientifold Dirac operator can be identifiedas a semi-equivariant Dixmier-Douady class. The main results are asfollows(a) Definition 31 defines ( Spin c , κ ǫ ) -structures . These are the appro-priate analogue of Spin c -strucutre for orientifolds.(b) Definition 36 defines the third integral orientifold Stiefel-Whineyclass , denoted W ( Γ ,ǫ ) .(c) Corollary 33 shows that W ( Γ ,ǫ ) is the obstruction to the existenceof a ( Spin c , κ ǫ ) -structure. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 3 (d) Corollary 34 shows that ( Spin c , κ ǫ ) -structure are classified by H Γ ( X , ( Z , ι ǫ )) , the elements of which correspond to semi-equivariantprincipal ( U ( ) , κ ǫ ) -bundles.(e) Theorem 41 provides an alternative criteria for the existence of a ( Spin c , κ ǫ ) -structure P → F r ( V ) based on the existence of a semi-equivariant ( U ( ) , κ ǫ ) -bundle that compliments the equivariantframe bundle Fr ( V ) .(f) Definition 54 defines the (Clifford-linear) orientifold Dirac operator and reduced orientifold Dirac operator .These results yeild the primary theorem of this paper. Theorem (57) . Let X be an orientifold with orientifold group ( Γ , ǫ ) .(a) If W ( Γ ,ǫ ) ( X ) (cid:3) , then X carries an orientifold Dirac operator.(b) If W ( Γ ,ǫ ) ( X ) (cid:3) and dim ( X ) (cid:3) , then X carries a reduced orientifoldDirac operator.In particular, if X is an -dimensional Real manifold and W ( Z , id ) ( X ) (cid:3) ,then X carries a reduced Real Dirac operator. These results appeared originally in the authors thesis [20]. Therelationship between the constructions described in this paper andother work in the literature will be discussed in §4, along with somepotential applications. Orientifolds
This section begins with a discussion of orientifold groups. Orien-tifold groups are topological groups equipped with a small amountof extra structure that allows them to act in a linear/anti-linear man-ner. The representation theory of such actions on finite dimensionalcomplex vector spaces can be reduced to the theory of unitary repre-sentations that are invariant under a conjugate structure on the spaceof equivalence classes of representations. This reduction is achieved byusing the Wigner’s notion of a corepresentation [29, pp. 334-335] [16,pp. 169-172], which coincides precisely with that of a semi-equivariant ( U ( n ) , κ ǫ ) -valued transition cocycle [21, §3] over a point.After briefly defining orientifolds , orientifold bundles will be intro-duced as complex vector bundles equipped with linear/anti-linear ac-tions. On any orientifold bundle, it is possible to construct a hermitianmetric that is compatible with the linear/anti-linear action. Moreover, SIMON KITSON the frame bundle of an orientifold bundle is a Γ -semi-equivariant prin-cipal ( U ( n ) , κ ǫ ) -bundle. Thus, a neat generalisation is formed, in whichan orientifold bundle over a point is an orientifold representation, andthe semi-equivariant ( U ( n ) , κ ǫ ) -valued transition cocycle of its framebundle is the corresponding corepresentation. From this perspective,the semi-equivariant transition cocycles defined in [21, §3] are gen-eralised corepresentations. As with equivariant bundles, orientifoldbundles admit a number of natural operations. Semi-equivariant co-cycles again prove useful, in §2.4, for defining and working with theseoperations.2.1. Orientifold Groups.
Any group Γ which acts by a combinationof linear and anti-linear operators must have an index- subgroup ofelements which act via linear operators, and a complementary subsetof elements which act via anti-linear operators. In general, if Γ con-tains more than one subgroup of index , then the set of orientifoldrepresentations of Γ depends on which of these groups is chosen as Γ + .These facts motivate the definition of an orientifold group. Definition 1. An orientifold group ( Γ , ǫ ) is a Lie group equipped witha non-trivial homomorphism ǫ : Γ → Z . For any orientifold groupdefine Γ + : (cid:3) k er ( ǫ ) and Γ − : (cid:3) Γ \ ker ( ǫ ) . Definition 2. A homomorphism ϕ : ( Γ ′ , ǫ ′ ) → ( Γ , ǫ ) of orientifold groupsis a group homomorphism such that ǫ ◦ ϕ (cid:3) ǫ ′ .The next lemma collects some basic facts about orientifold groups. Lemma 3. If ( Γ , ǫ ) is an orientifold group, then(a) Γ + ⊂ Γ is a normal subgroup(b) Γ / Γ + ≃ Z (c) → Γ + → Γ ǫ → Z → is an extension of topological groups(d) γ ∈ Γ + for all γ ∈ Γ (e) Γ (cid:3) Γ + ⊔ Γ − (cid:3) Γ + ⊔ ζ Γ + for any ζ ∈ Γ − . The simplest non-trivial example of an orientifold group is providedby id : Z → Z . Given an orientifold group, its semi-direct productwith a Γ -group can yield another orientifold group. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 5
Lemma 4.
Let ǫ : Γ → Z be an orientifold group and ( G , θ ) be a Γ -group.Then the group extension / / Γ + ⋉ θ G i / / Γ ⋉ θ G ǫ ◦ π / / Z / / ( γ, g ) ✤ / / ( γ, g ) ✤ / / ǫ ( γ ) makes Γ ⋉ θ G into an orientifold group. The notation ( Γ , ǫ ) ⋉ θ G will be usedto denote orientifold groups of this form. The following example commonly arises when G is a group of linearoperators and κ represents conjugation with respect to a fixed basis. Example 5.
Let ( G , θ ) be a Z -group with unit e , then ( Z , id ) ⋉ θ G isan orientifold group / / G i / / Z ⋉ θ G id ◦ π / / Z / / g ✤ / / ( z , g ) ✤ / / z . Note that the element (− , e ) ∈ Γ − is an involution, (− , e ) (cid:3) ((− ) , e (− e )) (cid:3) (− , e ) (cid:3) ( + , e ) . It is also possible to construct examples in which Γ − does not containan involution. Example 6.
The Weil group [1, §XV] of R is the subgroup C × ⊔ C × j ⊂ H × of the multiplicative group of quaternions. It fits into the non-splitextension / / C × / / C × ⊔ C × j / / Gal ( C / R ) / / j ✤ / / − of C × by Gal ( C / R ) ≃ Z , making it into an orientifold group. Note thatthere is no element ζ ∈ C × j (cid:3) Γ − such that ζ (cid:3) . Example 7. If H : (cid:3) {± , ± i } is the orientifold group equipped withthe homomorphism q ( h ) : (cid:3) h , then Γ : (cid:3) ( H , q ) ⋉ θ G is the orientifoldgroup / / {± } ⋉ θ G i / / {± , ± i } ⋉ θ G q ◦ π / / Z / / ( h , g ) ✤ / / ( h , g ) ✤ / / h . If ( h , g ) ∈ Γ − , then h (cid:3) ± i and ( h , g ) (cid:3) ( h , g ( h g )) (cid:3) (− , g ( h g )) ∈ Γ + .Thus, there is no element γ ∈ Γ − such that γ (cid:3) ( , e ) . SIMON KITSON
Given an orientifold group ( Γ , ǫ ) , the parity information providedby ǫ can be used when defining actions on various objects. Threedifferent types of actions of an orientifold group will be distinguished.The first type of action uses the parity information assigned to groupelements to dictate whether an element acts linearly or anti-linearly. Itwill be neccesary to define these actions on a variety of C -modules fromdifferent categories, including complex vector spaces, complex vectorbundles, and algebras over C . Given objects X and Y in an appropriatecategory, define Hom + ( X , Y ) : (cid:3) Hom ( X , Y ) Hom − ( X , Y ) : (cid:3) (cid:8) a ¯ Y ◦ ϕ | ϕ ∈ Hom ( X , ¯ Y ) (cid:9) , where a ¯ Y : ¯ Y → Y is the identity map on the underlying set for Y .The map a ¯ Y is anti-linear and the elements of Hom − ( X , Y ) can be con-sidered as anti-linear homomorphisms. The conjugation map Y ¯ Y changes the C -module structure of Y to its conjugate C -module struc-ture, and, depending on the category, it may change other structureson Y . For example, the conjugate of a Hilbert space carries a conju-gate inner product. Denote the disjoint union of Hom + and Hom − by Hom ± . The spaces End ± and A ut ± are defined similarly. Definition 8.
Let ( Γ , ǫ ) be an orientifold group. An orientifold action isa homomorphism ρ : Γ → Aut ± ( W ) such that ρ ( γ ) ∈ Aut ǫ ( γ ) ( W ) . A second type of action uses an involution ρ to define an actionof Γ . Typically, an involution of this type represents the change ofsome structure to a conjugate structure, occuring in parallel with theapplication of an orientifold action. Definition 9. An involutive action of an orientifold group, is an actionof the form ρ ◦ ǫ : Γ → Z → Aut ( Y ) , (1)where ρ : Z → Aut ( Y ) is an involution. Example 10.
Some examples of involutive actions are(a) ι p , q ǫ : R p , q → R p , q , where R p , q : (cid:3) R p ⊕ R q , ι p , q : ( x , y ) 7→ ( x , − y ) .(b) κ ǫ : GL ( n , C ) → GL ( n , C ) , where κ ǫ is elementwise conjugationon the standard matrix representation of GL ( n , C ) . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 7 (c) d θ ǫ : g → g , where g is a Lie algebra and θ : G → G is aninvolution on its Lie group.Throughout this paper, it will be assumed that C is equipped withthe orientifold action κ ǫ .Of course, the parity of the group elements can also be ignored.This type of action occurs on an orientifold and its tangent bundle. Inorder to differentiate it from the other types of action, it will be referedto as a basic action.2.2. Orientifolds.
In order to maintain a clear focus on anti-linearsymmetry, only the simplest definition of an orientifold will be treated.These orientifolds are essentially global quotient orbifolds with a smallamount of extra structure. Using the language of §2.1, they could bedescribed as manifolds equipped with a basic action of an orientifoldgroup. The origin of the term orientifold is in string theory, whereorientifolds are often considered to have a sign choices ± Definition 11. An orientifold is a compact manifold X equipped with asmooth action ρ : Γ → Diff ( X ) , where Γ is a finite orientifold group. The category of orientifolds withorientifold group ǫ : Γ → Z will be denoted Ori ( Γ ,ǫ ) . Example 12.
Let Γ be any orientifold group. Then R p , q : (cid:3) R p ⊕ R q equipped with the involutive action induced by ( x , y ) 7→ ( x , − y ) isan orientifold. This orientifold will be used to form suspensions inorientifold K -theory. Example 13.
Let X ∈ Ori ( Γ ,ǫ ) with Γ -action σ . The tangent bundle TX equipped with the basic Γ -action d σ is again an orientifold. The K -theory of this orientifold is the target space of the 8-fold Thom iso-morphism in orientifold K -theory.The category of real vector bundles equipped with a basic action ofthe orientifold group ( Γ , ǫ ) will be denoted Vect ( Γ ,ǫ ) ( X , R ) . The isomor-phism classes of such bundles will be denoted Vect ≃( Γ ,ǫ ) ( X , R ) .2.3. Orientifold Bundles.
Orientifold bundles are the main object ofinterest in the study of orientifolds. In the language of §2.1, theyare complex vector bundles carrying orientifold actions that cover theaction on the base orientifold.
SIMON KITSON
Definition 14. If π : E → X is a complex vector bundle, defineAut Diff ( E ) to be the set of maps ϕ : E → E such that(a) π ◦ ϕ ( e ) (cid:3) f ◦ π ( e ) , for some f ∈ Diff ( X ) and all e ∈ E .(b) ϕ : E x → E f ( x ) is a linear bijection, for all x . Definition 15. An orientifold bundle π : E → X is a complex vectorbundle equipped with an orientifold action τ : Γ → Aut ± Diff ( E ) such that π ( γ v ) (cid:3) γπ ( v ) .The category of orientifold bundles over X ∈ Ori ( Γ ,ǫ ) will be denotedVect ( Γ ,ǫ ) ( X , C ) . The set of isomorphism classes of orientifold bundleswill be denoted Vect ≃( Γ ,ǫ ) ( X , C ) . Example 16.
A linear/anti-linear representation ( V , ρ ) is an orientifoldbundle over a point. Such a bundle will also be refered to as an orien-tifold representation . If ( X , σ ) is an orientifold and ( V , ρ ) is an orientifoldrepresentation, then an orientifold bundle of the form ( X × V , σ × ρ ) will be described as a trivial orientifold bundle. Note that if ǫ is non-trivial, then every orientifold bundle for ( Γ , ǫ ) carries at least one anti-linear map, and so there is no orientifold bundle ( E , τ ) such that τ γ (cid:3) idfor all γ ∈ Γ .Just as in the equivariant setting, it is possible to average an her-metian metric on an orientifold bundle to make it compatible with theorientifold action. The averaging process needs to be twisted withconjugation to account for the anti-linearity of the action, as does thecompatibility condition. Definition 17. An orientifold metric on an orientifold bundle E is anhermitian metric h on E such that, for all v , v ∈ E and γ ∈ Γ , h ( γ v , γ v ) γ x (cid:3) γ h ( v , v ) x . Proposition 18.
Every orientifold vector bundle E over a paracompact ori-entifold X carries an orientifold metric. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 9
Proof.
It is a standard result that every complex vector bundle over aparacompact space carries an hermitian metric [27, Lemma 2]. Givenan hermitian metric h on an orientifold bundle E , define h Γ ( u , v ) x (cid:3) Õ γ ∈ Γ γ − h ( γ u , γ v ) γ x . This metric is an orientifold metric as h Γ ( γ u , γ v ) γ x (cid:3) Õ γ ′ ∈ Γ γ ′− h ( γ ′ γ u , γ ′ γ v ) γ ′ γ x (cid:3) Õ γ ′′ : (cid:3) γ ′ γ ∈ Γ γγ ′′− h ( γ ′′ u , γ ′′ v ) γ ′′ x (cid:3) γ Õ γ ′′ ∈ Γ γ ′′− h ( γ ′′ u , γ ′′ v ) γ ′′ x (cid:3) γ h Γ ( u , v ) x . Using an orientifold metric, it is possible to split sequences of ori-entifold bundles.
Corollary 19.
Let X be a paracompact orientifold. If → E ′ ϕ ′ → E ϕ → E ′′ is an exact sequence of orientifold bundles over X , then E ≃ E ′ ⊕ E ′′ .Proof. By Proposition 18, there exists an orientifold metric h on E . Itis a standard result [27, Proposition 2] that h determines a projection p : E → E and a splitting of complex vector bundles E (cid:3) im ( p ) ⊕ ker ( p ) ≃ E ′ ⊕ E ′′ . The projection p is defined fibrewise by p x : E x → E x v Õ i h ( v , b i ) x h ( b i , b i ) x b i , where { b i } is any basis for ϕ ′ ( E ′ ) x . Therefore, if p x ( v ) (cid:3)
0, then h ( v , b i ) x (cid:3) i , and p γ x ( γ v ) (cid:3) Õ i h ( γ v , γ b i ) γ x h ( γ b i , γ b i ) γ x ( γ b i ) (cid:3) Õ i γ h ( v , b i ) x γ h ( b i , b i ) x ( γ b i ) (cid:3) Õ i γ γ h ( b i , b i ) x ( γ b i ) (cid:3) . Thus, ker ( p ) is invariant under the action of Γ , as is the given splitting. Next, the frame bundle of an orientifold bundle will be examined.
Definition 20.
The frame bundle Fr ( E ) of an orientifold bundle E is theprincipal GL ( n , C ) -bundle of frames for the total space of E , equippedwith a left Γ -action defined on a frame s (cid:3) ( s , . . . , s n ) ∈ Fr ( E ) x by ( γ s ) i (cid:3) γ s i .Although the frame bundle of an orientifold is defined in the samemanner as that of an equivariant bundle, the anti-linearity present inthe Γ -action gives it different properties. In particular, there is a mildnoncommutivity between the left action of Γ and the right action ofthe structure group GL ( n , C ) . This non-commutivity makes the framebundle of an orientifold bundle into a Γ -semi-equivariant principal ( GL ( n , C ) , κ ǫ ) -bundle [21, §2]. Proposition 21.
Let E be an orientifold bundle and consider GL ( n , C ) to beequipped with the involutive action of ( Γ , ǫ ) induced by conjugation. Then, Fr ( E ; GL ( n , C )) ∈ PB Γ ( X , ( GL ( n , C ) , κ ǫ )) . In particular, the left and right actions on the frame bundle satisfy γ ( s g ) (cid:3) ( γ s )( γ g ) , for γ ∈ Γ , s ∈ Fr ( E ) and g ∈ GL ( n , C ) .Proof. The action of g on a frame s is given by ( s g ) j (cid:3) Í ≤ i ≤ n s i g ij .Thus, γ ( s g ) j (cid:3) Õ ≤ i ≤ n γ ( s i g ij ) (cid:3) Õ ≤ i ≤ n ( γ s i )( γ g ij ) (cid:3) Õ ≤ i ≤ n ( γ s ) i ( γ g ) ij (cid:3) (( γ s )( γ g )) j . Note that, by using an orientifold metric, the structure group canalways be reduced to ( U ( n ) , κ ǫ ) , where κ ǫ is the action induced on U ( n ) by its inclusion into GL ( n , C ) .2.4. Operations on Orientifold Bundles.
Some basic operations onorientifold bundles will now be defined. It will be useful to make thesedefinitions in terms of semi-equivariant cocycles [21, §3]. To start with,consider the following operations on Γ -groups. Definition 22.
Let a k ∈ GL ( C m k ) , and denote by [ a ij ] the matrix repre-sentation of an element a ∈ GL ( C m ) with respect to the standard basisof C m . Define the following operations HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 11 (a) The dual a ∗ ∈ GL ( C m ) , [( a ∗ ) ij ] : (cid:3) ([ a ij ] t ) − (b) The direct sum a ⊕ a ∈ GL ( C m + m ) , [( a ⊕ a ) ij ] : (cid:3) [ a ij ] [ a ij ] ! . (c) The tensor product a ⊗ a ∈ GL ( C m m ) , [( a ⊗ a ) ij ] : (cid:3) © « a [ a ij ] . . . a m [ a ij ] ... . . . ... a m [ a ij ] . . . a mm [ a ij ] ª®®¬ . Examining Definition 22, it is clear that the dual, direct sum andtensor product on the groups GL ( C m ) are compatible with involutive Γ -actions induced by conjugation. Lemma 23.
The dual, direct sum and tensor product operations are homo-morphisms ∗ : ( GL ( C m ) , κ ǫ ) → ( GL ( C m ) , κ ǫ )⊕ : ( GL ( C m ) , κ ǫ ) × ( GL ( C m ) , κ ǫ ) → ( GL ( C m + m ) , κ ǫ )⊗ : ( GL ( C m ) , κ ǫ ) × ( GL ( C m ) , κ ǫ ) → ( GL ( C m m ) , κ ǫ ) of Γ -groups. Lemma 23, allows the dual, direct sum and tensor product of ( GL ( m , C ) , κ ǫ ) -valued transition cocycles [21, §3] to be defined in theobvious way. Pullbacks of cocycles can also be defined. It is routine toprove that these satisfy the semi-equivariant cocycle condition. Definition 24.
Let φ i ∈ TC ( Γ ,ǫ ) (U , X , ( GL ( C m i ) , κ ǫ )) and f : X → Y bea homomorphism orientifolds. The pullback , dual , direct sum , and tensorproduct are defined, respectively, by ( f ∗ φ ) ba ( γ, x ) : (cid:3) φ ba ( γ, f ( x )) ∈ TC ( Γ ,ǫ ) ( f ∗ U , Y , ( GL ( C m ) , κ ǫ ))( φ ∗ ) ba ( x , γ ) : (cid:3) φ ba ( x , γ ) ∗ ∈ TC ( Γ ,ǫ ) (U , X , ( GL ( C m ) , κ ǫ ))( φ ⊕ φ ) ba ( x , γ ) : (cid:3) φ ba ( x , γ ) ⊕ φ ba ( x , γ ) ∈ TC ( Γ ,ǫ ) (U , X , ( GL ( C m + m ) , κ ǫ ))( φ ⊗ φ ) ba ( x , γ ) : (cid:3) φ ba ( x , γ ) ⊗ φ ba ( x , γ ) ∈ TC ( Γ ,ǫ ) (U , X , ( GL ( C m m ) , κ ǫ )) , where f ∗ U : (cid:3) (cid:8) f − ( U a ) | a ∈ A (cid:9) is the pullback of U : (cid:3) { U a | a ∈ A } . The above operations on cocycles induce operations on orientifoldbundles via the semi-equivariant associated bundle construction, seeDefinition 59.
Definition 25.
Let E i ∈ Vect m i ( Γ ,ǫ ) ( X , C ) . Let φ i denote a semi-equivariantcocycle associated Fr ( E i ) by [21, Prop. 12], and P φ denote the semi-equivariant principal bundle constructed from a cocycle φ via [21,Prop. 15]. P φ denote The pullback , dual , direct sum , and tensor product operations on orientifold bundles are defined, respectively, by f ∗ E : (cid:3) P f ∗ φ × ( GL ( m , C ) ,κ ǫ ) ( C m , κ ǫ ) ∈ Vect m ( Γ ,ǫ ) ( X , C ) E ∗ : (cid:3) P φ ∗ × ( GL ( m , C ) ,κ ǫ ) (( C m ) ∗ , κ ǫ ) ∈ Vect m ( Γ ,ǫ ) ( X , C ) E ⊕ E : (cid:3) P φ ⊕ φ × ( GL ( m + m , C ) ,κ ǫ ) ( C m + m , κ ǫ ) ∈ Vect m + m ( Γ ,ǫ ) ( X , C ) E ⊗ E : (cid:3) P φ ⊗ φ × ( GL ( m m , C ) ,κ ǫ ) ( C m m , κ ǫ ) ∈ Vect m m ( Γ ,ǫ ) ( X , C ) , where κ ǫ : ( C m ) ∗ → ( C m ) ∗ is the action defined by ( γλ )( z ) : (cid:3) γλ ( γ − z ) .As in the non-equivariant setting, it is possible to construct the bun-dle of homomorphisms between two orientifold bundles using theirtensor products and duals. Proposition 26.
Let E i ∈ Vect m i ( Γ ,ǫ ) ( X , C ) . Homomorphisms in Hom ( E , E ) correspond bijectively to equivariant sections of the orientifold bundle E ⊗ E ∗ . Classification of
Spin c -Structures on Orientifolds. In order todefine and classify Spin c -structures for orientifolds, it is neccesary toconsider the interaction of Clifford algebras and the Spin groups withorientifold actions. The idea is to complexify results which apply to realClifford algebras, whilst keeping track of the associated conjugationmaps. These maps can then be used to define involutive actions oforientifold groups. To begin, the definitions of the real Clifford algebra,Spin group, and adjoint map are recalled. Definition 27.
The Clifford algebra Cl n is the algebra generated bythe standard basis { e i } of R n subject to the relations e i (cid:3) − e i e j + e j e i (cid:3) (cid:8) e i · · · e i k ∈ Cl n | i < · · · < i k (cid:9) is a basis for Cl n .The group Spin ( n ) sits inside Cl n . Elements of Spin ( n ) are products ofan even number of unit vectors from R n . Definition 28.
The group Spin ( n ) is defined bySpin ( n ) : (cid:3) { x · · · x k | x i ∈ R n , k x i k (cid:3) } ⊂ Cl n . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 13 If g ∈ Spin ( n ) and x ∈ R n one can show that gx g − ∈ R n . The trans-formation x gx g − defines an element of SO ( n ) , and the resultingassignment Spin ( n ) → SO ( n ) is a double covering. Definition 29.
The adjoint map
Ad : Spin ( n ) → SO ( n ) is defined, for g ∈ Spin ( n ) , x ∈ R n , by Ad g ( x ) : (cid:3) gx g − . For applications to orientifolds, it is neccesary to work with thecomplexifications of Cl n and Spin ( n ) . These complexifications areequipped with conjugation maps which induce involutive actions oforientifold groups. The complexified adjoint map is a homomorphismof Γ -groups. Definition 30.
Let ( Γ , ǫ ) be an orientifold group and define the follow-ing(a) ( C l n , κ ǫ ) : (cid:3) Cl n ⊗ C with the Γ -action κ ǫ ( ϕ ⊗ z ) : (cid:3) ϕ ⊗ κ ǫ ( z ) (b) ( Spin c ( n ) , κ ǫ ) : (cid:3) ( Spin ( n ) × U ( ))/{±( , )} with the induced ac-tion κ ǫ [ g , z ] : (cid:3) [ g , κ ǫ ( z )] (c) Ad c : ( Spin c ( n ) , κ ǫ ) → ( SO ( n ) , id ǫ ) defined by Ad c [ g , z ] : (cid:3) Ad ( g ) .Note that Ad c ◦ κ ǫ [ g , z ] (cid:3) Ad c [ g , z ] . The properties of Ad c , andthe decomposition of Spin c ( n ) , produce two central exact sequencesof Γ -groups about Spin c ( n ) . These sequences fit into the followingdiagram ( Spin ( n ) , id ǫ ) ( Spin c ( n ) , κ ǫ ) ( U ( ) , κ ǫ ) ( SO ( n ) , id ǫ ) Ad c Ad ( U ( ) , κ ǫ ) q ( Z , id ǫ ) ( Z , id ǫ ) , (2) where q is the square map. The above sequences will be used to classifySpin c -structures for orientifolds.Having examined semi-equivariance, orientifolds, and orientifoldactions on Spin c ( n ) , it is now possible to define a notion of Spin c -structure which is appropriate for orientifolds. Definition 31. An ( Spin c , κ ǫ ) -structure for a real Γ -equivariant vectorbundle V over an orientifold is a semi-equivariant lifting ϕ : P → Fr ( V ) by Ad c : ( Spin c ( n ) , κ ǫ ) → ( SO ( n ) , id ǫ ) .If V has a ( Spin c , κ ǫ ) -structure, then it is said to be ( Spin c , κ ǫ ) -oriented . If the tangent bundle T M of an orientifold M is ( Spin c , κ ǫ ) -oriented, then M is said to be ( Spin c , κ ǫ ) - oriented .The ( Spin c , κ ǫ ) -structures associated to a vector bundle V can beclassified using the results of [21]. The following theorem is obtainedby applying [21, Theorem 41] to the central exact sequence runningvertically in diagram (2). Theorem 32.
The central exact sequence → ( U ( ) , κ ǫ ) → ( Spin c ( n ) , κ ǫ ) Ad c → ( SO ( n ) , id ǫ ) → , induces an exact sequence H Γ ( X , ( U ( ) , κ ǫ )) → TC Γ ( X , ( Spin c ( n ) , κ ǫ )) Ad c → . . .. . . Ad c → TC Γ ( X , ( SO ( n ) , id ǫ )) ∆ sc → H Γ ( X , ( U ( ) , κ ǫ )) . Theorem 32 has the following corollaries, which classify ( Spin c , κ ǫ ) -structures in terms of semi-equivariant cohomology with coefficientsin ( U ( ) , κ ǫ ) . Corollary 33.
A real Γ -equivariant vector bundle V over an orientifold has a ( Spin c , κ ǫ ) -structure if and only if ∆ sc ( φ V ) (cid:3) , where φ V is the transitioncocycle for V . Corollary 34.
A given ( Spin c , κ ǫ ) -structure is unique up to tensoring bysemi-equivariant principal ( U ( ) , κ ǫ ) -bundles. To obtain an obstruction class with integer coefficients, involutiveactions can be taken on the groups in the exponential exact sequence.This results in the following proposition.
Lemma 35.
The exponential exact sequence → ( Z , ι ǫ ) → ( R , ι ǫ ) exp → ( U ( ) , κ ǫ ) → HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 15 induces isomorphisms H p Γ ( X , ( U ( ) , κ ǫ )) ∆ p exp → H p + Γ ( X , ( Z , ι ǫ )) , where ι ǫ is the involutive orientifold action induced by the map t
7→ − t ∈ R .Proof. By [21, Theorem 38], the exact sequence (3) induces a long exactsequence H p Γ ( X , ( Z , ι ǫ )) → H p Γ ( X , ( R , ι ǫ )) exp → H p Γ ( X , ( U ( ) , κ ǫ )) ∆ p exp → H p + Γ ( X , ( Z , ι ǫ )) . The cohomology groups H p Γ ( X , ( R , ι ǫ )) vanish for all p , due to the ex-istence of a smooth partition of unity on X . Therefore, the maps ∆ p exp are isomorphisms.Using Lemma 35, it is possible to define an analogue of the thirdintegral Stiefel-Whiney class. Definition 36.
The third integral orientifold Stiefel-Whiney class is definedby W ( Γ ,ǫ ) ( V ) : (cid:3) ∆ exp ◦ ∆ sc ( φ V ) ∈ H Γ ( X , ( Z , ι ǫ )) , where φ V is the transition cocycle associated to V .Corollaries 33 and 34 can then be restated in terms of semi-equivariantcohomology with coefficients in ( Z , ι ǫ ) . Corollary 37.
A real Γ -equivariant bundle V is ( Spin c , κ ǫ ) -oriented if andonly if W ( Γ ,ǫ ) ( V ) (cid:3) . Corollary 38.
The ( Spin c , κ ǫ ) -structures on a ( Spin c , κ ǫ ) -oriented real Γ -equivariant vector bundle are in bijective correspondence with the elements of H Γ ( X , ( Z , ι ǫ )) . It is possible to further isolate the semi-equivariance in a ( Spin c , κ ǫ ) -structure by splitting it via the decomposition ( Spin c ( n ) , κ ) ≃ ( SO ( n ) , id ) × Z ( U ( ) , κ ) . This decomposition immediately implies that, for any cochain φ sc ∈ K Γ ( X , ( Spin c ( n ) , κ ǫ )) , there exist cochains φ s ∈ K Γ ( X , ( Spin ( n ) , id ǫ )) and φ u ∈ K Γ ( X , ( U ( ) , κ ǫ )) such that φ sc (cid:3) [ φ s , φ u ] . It also allows thedefinition of the mapAd × q : ( Spin c ( n ) , κ ǫ ) → ( SO ( n ) , id ǫ ) × ( U ( ) , κ ǫ )[ s , z ] 7→ ( Ad ( s ) , q ( z )) . The next proposition shows that every ( Spin c , κ ǫ ) -structure extends toa lifting of a semi-equivariant principal ( SO ( n ) , id ) × ( U ( ) , κ ) -bundleby Ad × q . Proposition 39. If ϕ : P → Q is a ( Spin c , κ ǫ ) -structure, then there existsa lifting ϕ : P → Q × X L (4) by Ad × q , where L is a Γ -semi-equivariant principal ( U ( ) , κ ǫ ) -bundle.Proof. Let φ ∈ TC Γ ( X , ( SO ( n ) , id ǫ )) be the cocycle for Q . If Q has a ( Spin c , κ ǫ ) -structure there is a cocycle [ φ s , φ u ] ∈ TC Γ ( X , ( Spin c ( n ) , κ ǫ )) with Ad c ([ φ s , φ u ]) (cid:3) Ad ( φ s ) (cid:3) φ . The cocycle [ φ s , φ u ] is a lifting byAd × q of ( φ, φ u ) . It remains to check that φ u is a cocycle. First, notethat Ad ( ∂φ s ) (cid:3) ∂ ◦ Ad ( φ s ) (cid:3) ∂ ( φ ) (cid:3)
1. Thus, ∂φ s takes values inker ( Ad ) (cid:3) Z , and ( ∂φ s ) − ( ∂φ u ) ∈ K Γ ( X , ( U ( ) , κ ǫ )) ⊂ K Γ ( X , ( Spin c ( n ) , κ ǫ )) . This cochain is a cocycle as ( ∂φ s ) − ( ∂φ u ) (cid:3) [ , ( ∂φ s ) − ( ∂φ u )] (cid:3) [ ∂φ s , ∂φ u ] (cid:3) ∂ [ φ s , φ u ] (cid:3) . The cochain φ u ∈ K Γ ( X , ( U ( ) , κ ǫ )) is then a cocycle as ∂ ( φ u ) (cid:3) ( ∂φ u ) (cid:3) ( ∂φ s ) − ( ∂φ u ) (cid:3) (cid:16) ( ∂φ s ) − ( ∂φ u ) (cid:17) (cid:3) . Therefore, the required bundle L can be constructed from φ u using [21,Prop. 15].Proposition 39 can be refined into a statement about cohomologyclasses. This refinement uses the exact sequences in cohomology ob-tained by applying [21, Theorem 41] to the two exact sequences of Γ -groups running diagonally in diagram (2). Lemma 40.
The central exact sequences → ( Z , id ǫ ) → ( Spin ( n ) , id ǫ ) A d → ( SO ( n ) , id ǫ ) → , → ( Z , id ǫ ) → ( U ( ) , κ ǫ ) q → ( U ( ) , κ ǫ ) → , induce the exact sequences H Γ ( X , ( Z , id ǫ )) → TC Γ ( X , ( Spin ( n ) , id ǫ )) Ad → . . .. . . Ad → TC Γ ( X , ( SO ( n ) , id ǫ )) ∆ s → H Γ ( X , ( Z , id ǫ )) , H Γ ( X , ( Z , id ǫ )) → H Γ ( X , ( U ( ) , κ ǫ )) q → . . .. . . q → H Γ ( X , ( U ( ) , κ ǫ )) ∆ u → H Γ ( X , ( Z , id ǫ )) . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 17
Proposition 39 and Lemma 40 can now be combined to establish analternative criteria for the existence of a ( Spin c , κ ǫ ) -structure. Theorem 41. A Γ -equivariant principal SO ( n ) -bundle Q with cocycle φ has a ( Spin c , κ ǫ ) -structure if and only if there exists a cocycle ψ ∈ H Γ ( X , ( U ( ) , κ ǫ )) such that ∆ s ( φ ) (cid:3) ∆ u ( ψ ) ∈ H Γ ( X , ( Z , id ǫ )) . Proof.
Assume that Q has a ( Spin c , κ ǫ ) -structure. By Proposition 39,there exists an cocycle [ φ s , φ u ] ∈ TC Γ ( X , ( Spin c ( n ) , κ ǫ )) such that φ u isa cocycle and ( Ad × q )[ φ s , φ u ] (cid:3) ( φ s , φ u ) . As [ φ s , φ u ] is a cocycle, [ ∂φ s , ∂φ u ] (cid:3) ∂ [ φ s , φ u ] (cid:3)
1. This implies that ∂φ s (cid:3) ∂φ u . Therefore, applying Lemma 40 to φ and φ u , ∆ s ( φ ) (cid:3) [ ∂φ s ] (cid:3) [ ∂φ u ] (cid:3) ∆ u ( φ u ) ∈ H Γ ( X , ( Z , id ǫ )) . Thus, ψ : (cid:3) φ u is the required cocycle.Conversely, suppose there exists a cocycle ψ ∈ H Γ ( X , ( U ( ) , κ ǫ )) such that ∆ s ( φ ) (cid:3) ∆ u ( ψ ) ∈ H Γ ( X , ( Z , id ǫ )) . Then, there are a cochains φ s with Ad ( φ s ) (cid:3) φ , and φ u with φ u (cid:3) ψ such that [ ∂φ s ] (cid:3) [ ∂φ u ] ∈ K Γ ( X , ( Z , id ǫ )) . This implies that ∂φ s (cid:3) ∂φ ′ ∂φ u (cid:3) ∂ ( φ ′ φ u ) for some φ ′ ∈ K Γ ( X , ( Z , id ǫ )) .Then ∂ [ φ s , φ ′ φ u ] (cid:3) [ ∂φ s , ∂ ( φ ′ φ u )] (cid:3)
1, and Ad c [ φ s , φ ′ φ u ] (cid:3) Ad ( φ s ) (cid:3) φ . Thus, [ φ s , φ ′ φ u ] defines a ( Spin c , κ ǫ ) -structure on Q .If X is a manifold acted on by a finite group H , and V → X is a real H -equivariant vector bundle with cocycle φ ∈ TC H ( X , SO ( n )) , then theobstruction to the existence of an H -equivariant Spin-structure on V is the second Z -valued equivariant Stiefel-Whitney class, which canbe defined by w H ( V ) : (cid:3) ∆ Spin ( φ ) ∈ H H ( X , Z ) . Here ∆ Spin ( φ ) is theconnecting map for the exact sequence H H ( X , Z ) / / TC H ( X , Spin ( n )) Ad / / TC H ( X , SO ( n )) ∆ Spin / / H H ( X , Z ) , induced by the central exact sequence1 → Z → Spin ( n ) Ad → SO ( n ) → . If ( Γ , ǫ ) is the orientifold group defined by Γ : (cid:3) Z × H and ǫ ( z , h ) : (cid:3) z , then X can be made into an orientifold ˜ X for ( Γ , ǫ ) by trivially extending its H -action to the Γ -action ( z , h ) x : (cid:3) hx . Similarly, the H -equivariant vector bundle V can be made into a Γ -equivariant vectorbundle ˜ V by trivially extending its H -action to the Γ -action ( z , h ) v : (cid:3) hv . The cocycle of ˜ V is an element ˜ φ ∈ TC Γ ( X , ( SO ( n ) , id ǫ )) .In this situation, the quotient map π : Γ → Γ / Z ≃ H induces a map π : X • Γ → X • H between the simplicial spaces associated to the groups Γ and H . Because π is a homomorphism and satisfies π ( γ ) x (cid:3) γ x , itcommutes with the face maps on these spaces, and defines a pulbackmap π ∗ on cochains. The map π ∗ also commutes with the coboundarymaps, and provides well-defined extension maps π ∗ : TC pH ( X , G ) → TC p Γ ( ˜ X , ( G , id ǫ )) π ∗ : H pH ( X , G ) → H p Γ ( ˜ X , ( G , id ǫ )) . One then has the following result.
Proposition 42. If ˜ V → ˜ X is the trivial extension of a real H -equivariantvector bundle V → X , as described above, then(a) the cocycle for ˜ V is the pullback of the cocycle for V by the quotient map π : Γ → H , ˜ φ (cid:3) π ∗ φ ∈ H Γ ( ˜ X , ( SO ( n ) , id ǫ )) . (b) the second Z -valued equivariant Stiefel-Whitney class for V satisfies π ∗ w H ( V ) (cid:3) ∆ s ( π ∗ φ ) ∈ H Γ ( ˜ X , ( Z , id ǫ )) . (c) ˜ V has a ( Spin c , κ ǫ ) -structure if and only if π ∗ w H ( V ) (cid:3) ∆ u ( ψ ) ∈ H Γ ( ˜ X , ( Z , id ǫ )) , for some cocycle ψ ∈ H Γ ( ˜ X , ( U ( ) , κ ǫ )) .Here ∆ s and ∆ u are the connecting maps of Lemma 40.Proof. If { s a } is a collection of local sections for V , then π ( z , h ) x (cid:3) hx (cid:3) ( z , h ) x π ( z , h ) s a ( x ) (cid:3) hs a ( x ) (cid:3) ( z , h ) s a ( x ) , where ( z , h ) ∈ Γ (cid:3) Z × H , x ∈ X . Together with the property whichdefines the cocycles φ and ˜ φ [21, Prop. 12], this implies s b ( π ( z , h ) x ) φ ba ( π ( z , h ) , x ) (cid:3) π ( z , h ) s a ( x ) (cid:3) ( z , h ) s a ( x ) (cid:3) s b (( z , h ) x ) ˜ φ ba (( z , h ) , x ) (cid:3) s b ( π ( z , h ) x ) ˜ φ ba (( z , h ) , x ) . Thus, π ∗ φ (cid:3) ˜ φ , which proves the the first statement. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 19
The second statement follows from the existence of the commutativediagram H H ( X , Z ) (cid:15) (cid:15) π ∗ / / H Γ ( ˜ X , ( Z , id ǫ )) (cid:15) (cid:15) TC H ( X , Spin ( n )) A d (cid:15) (cid:15) π ∗ / / TC Γ ( ˜ X , ( Spin ( n ) , id ǫ )) Ad (cid:15) (cid:15) TC H ( X , SO ( n )) ∆ Spin (cid:15) (cid:15) π ∗ / / TC Γ ( ˜ X , ( SO ( n ) , id ǫ )) ∆ s (cid:15) (cid:15) H H ( X , Z ) π ∗ / / H Γ ( ˜ X , ( Z , id ǫ )) . To see that the bottom cell of this diagram commutes, note that if ψ isa lifting of φ , then π ∗ ψ is a lifting of π ∗ φ . The commutation of π ∗ withthe coboundary maps then implies π ∗ w H ( V ) : (cid:3) π ∗ ∆ Spin ( φ ) (cid:3) π ∗ ∂ ( ψ ) (cid:3) ∂ ( π ∗ ψ ) (cid:3) ∆ s ( π ∗ φ ) . The third statement follows from the first and second by applyingTheorem 41.To end this section, two important ( Spin c , κ ǫ ) -structures will bedescribed. The first of these is the canonical ( Spin c , κ ǫ ) -structure as-sociated to a real representation V of ( Z , id ) ⋉ κ ǫ Spin c ( n ) . Whendim ( V ) (cid:3)
8, this ( Spin c , κ ǫ ) -structure is used to construct a canon-ical reduced orientifold spinor bundle over the point orientifold for ( Z , id ) ⋉ κ ǫ Spin c ( n ) , which, in turn, can be used to construct the 8-foldBott class over V for orientifold K -theory [20, Example 4.9]. The secondis a canonical ( Spin c , κ ǫ ) -structure on the n -sphere. This ( Spin c , κ ǫ ) -structure is used to construct a canonical reduced orientifold spinorbundle on S k . The reduced orientifold spinor bundle on S k can beused to describe the compactification of the 8-fold Bott class over a realrepresentation of ( Z , id ) ⋉ κ ǫ Spin c ( n ) [20, Example 4.11]. . Lemma 43 (The canonical ( Spin c , κ ǫ ) -structure over a point) . Let V be therepresentation of ( Z , id ) ⋉ κ ǫ Spin c ( n ) on R n defined by ( γ, g )· v : (cid:3) Ad c ( g ) v .Then Ad c : Spin c ( n ) → SO ( n ) ≃ Fr ( V ) . is a ( Spin c , κ ǫ ) -structure for the real equivariant vector bundle V → pt overthe point orientifold for ( Z , id ) ⋉ κ ǫ Spin c ( n ) . Proof.
The group Spin c ( n ) forms a principal bundle over a point withthe trivial projection π ( p ) (cid:3) pt, and right Spin c ( n ) action defined bymultiplication. The left action of ( Z , id ) ⋉ κ ǫ Spin c ( n ) is taken to be ( γ, g ) · p : (cid:3) g κ γ ( p ) , for γ ∈ Γ and g , p ∈ Spin c ( n ) . The inclusion of the conjugation κ isthe only difference from the corresponding construction in the usualequivariant setting. Lemma 44 (The canonical ( Spin c , κ ǫ ) -structure on the sphere) . The map Ad c : Spin c ( n + ) → SO ( n + ) forms a ( Spin c , κ ǫ ) -structure for the orientifold S n ⊂ R n + equipped with theaction of ( Z , id ) ⋉ κ ǫ Spin c ( n + ) defined by ( γ, g ) · v : (cid:3) Ad c ( g ) v .Proof. In what follows, let γ ∈ Z , g , p ∈ Spin c ( n + ) , h ∈ Spin c ( n ) , q ∈ SO ( n + ) , f ∈ SO ( n ) . Also, let α : SO ( n ) → SO ( n + ) and β : Spin c ( n ) → Spin c ( n + ) be the maps induced by the inclusion C l n → C l n + defined on standard basis elements by e k e k + . EquipSpin c ( n + ) with the projection, left action, and right Spin c ( n ) -action π sc ( p ) : (cid:3) Ad c ( p ) e ( γ, g ) · p : (cid:3) g κ γ ( p ) p · h : (cid:3) p β ( h ) , respectively. Again, the presence of the conjugation action κ in theleft action is the only difference from the corresponding constructionin the usual equivariant setting [7, p. 5]. Using the properties of κ ,Ad c and β , it is straightforward to check that Spin c ( n + ) forms a ( Γ , ǫ ) ⋉ κ ǫ Spin c ( n + ) -semi-equivariant principal ( Spin c ( n ) , κ ǫ ) -bundle, π sc (( γ, g ) · p ) (cid:3) π ( g ( γ p )) (cid:3) Ad c ( g ( γ p )) e (cid:3) Ad c ( g ) Ad c ( γ p ) e (cid:3) Ad c ( g ) Ad c ( p ) e (cid:3) ( γ, g ) π sc ( p ) , ( γ, g ) · ( p · h ) (cid:3) ( γ, g ) · ( p β ( h )) (cid:3) g ( γ ( p β ( h ))) (cid:3) g ( γ p )( γβ ( h )) (cid:3) g ( γ p ) β ( γ h ) (cid:3) (( γ, g ) p ) · ( γ h ) . Next, equip SO ( n + ) with the projection, left action, and right SO ( n ) -action defined by π so ( q ) : (cid:3) qe ( γ, g ) · q : (cid:3) Ad c ( g ) q q · f : (cid:3) q α ( f ) , HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 21 respectively. It can then be checked that SO ( n + ) forms a ( Z , id ) ⋉ κ ǫ Spin c ( n + ) -equivariant principal SO ( n ) -bundle, π so (( γ, g ) · q ) (cid:3) π so ( Ad c ( g ) q ) (cid:3) Ad c ( g ) qe (cid:3) ( γ, g ) π ( q ) , ( γ, g ) · ( q · f ) (cid:3) ( γ, g ) · ( q α ( f )) (cid:3) Ad ( g ) q α ( f ) (cid:3) (( γ, g ) · q ) · f . That Ad c is a semi-equivariant lifting can be checked directly by veri-fying compatibility with projections, right actions, and left actions, π sc ( p ) (cid:3) Ad c ( p ) e (cid:3) π so ◦ Ad c ( p ) , Ad c ( p · h ) (cid:3) Ad c ( p β ( h )) (cid:3) Ad c ( p ) Ad c ( β ( h )) (cid:3) Ad c ( p ) α ( Ad c ( h )) (cid:3) Ad c ( p ) · Ad c ( h ) , Ad c (( γ, g ) · p ) (cid:3) Ad c ( g ( γ p )) (cid:3) Ad c ( g ) Ad c ( γ p ) (cid:3) Ad c ( g ) Ad c ( p ) (cid:3) ( γ, g ) · Ad c ( p ) . It remains to check that SO ( n + ) with the given action of ( Z , id ) ⋉ κ ǫ Spin c ( n + ) is isomorphic to the equivariant principal SO ( n ) -bundleFr ( S n ) . First, identify the tangent space of the n -sphere with a subbun-dle of the tangent space to R n + , TS n ≃ (cid:8) ( v , v ) ∈ R n + × R n + | k v k (cid:3) k v k (cid:3) , h v , v i (cid:3) (cid:9) ⊂ T R n + The standard action of SO ( n + ) on R n + associates a matrix to eachelement q ∈ SO ( n + ) , which will also be denoted q . The columns q i of this matrix determine an orthonormal frame F ( q ) : (cid:3) (cid:8) ( q , q ) , . . . , ( q , q n + ) (cid:9) ∈ Fr q ( TS n ) . In this way, SO ( n + ) can be identified with Fr ( TS n ) . This identificationis compatible with projections as π so ( q ) (cid:3) qe (cid:3) q (cid:3) π TS n ( F ( q )) . Compatibility with right actions follows from the fact that ( q · f ) j (cid:3) ( q α ( f )) j (cid:3) ( q for j (cid:3) Í ≤ i ≤ n + q i f ( i − )( j − ) for j ≥ . Finally, the left action on Fr ( TS n ) can be characterised by observingthat a vector ( v , v ) ∈ TS n is tangent to the curve ( cos t ) v + ( sin t ) v at t (cid:3)
0. Acting on this curve by ( γ, g ) ∈ ( Z , id ) ⋉ κ ǫ Spin c ( n + ) produces a new curve ( cos t )( Ad c ( g ) v ) + ( sin t )( Ad c ( g ) v ) which has ( Ad c ( g ) v , Ad c ( g ) v ) as its tangent vector at t (cid:3)
0. Thus, ( γ, g ) F ( q ) (cid:3) F ( Ad c ( g ) q ) (cid:3) F (( γ, g ) q ) , and the identification of SO ( n + ) and Fr ( TS n ) is compatible with theleft actions.
3. Dira c Operators on Orientifolds
In this section, Dirac operators are constructed for orientifolds. By ap-plying a semi-equivariant associated bundle construction with a Clif-ford module as the model fibre, it is possible to construct spinor bun-dles with orientifold actions. Both a total spinor bundle, with a rightaction of ( C l n , κ ǫ ) , and a reduced spinor bundle, with the complexi-fication of an irreducible Cl k -module as a model fibre, are defined.As in the usual setting, the sections of orientifold spinor bundles areacted on by sections of a Clifford bundle. This action is compatiblewith the orientifold action on the spinor bundle and a canonical ori-entifold action on the complex Clifford bundle. In order to constructa Dirac operator on an orientifold, it is neccesary to have a connectionwhich is compatible with Clifford multiplication on sections and theorientifold action. Such a connection can be constructed using resultson semi-equivariant connection forms from §A.2. After equipping theorientifold spinor bundles with compatible connections, the orientifoldDirac operator and its reduced counterpart will be defined.3.1. Orientifold Spinor Bundles.
The model fibre of an orientifoldspinor bundle is a Clifford module that is semi-equivariant with respectto the action of ( Spin c ( n ) , κ ǫ ) . Such modules can be constructed bycomplexifying Cl n -modules. The main Cl n -modules of interest are Cl n ,considered as a module over itself, and the irreducible Cl k -modules.Up to equivalence, there is only one irreducible Cl k -module [23, p. 33].A representative of this equivalence class will be denoted by ∆ . De-note the complexifications of these, equipped with their associatedorientifold actions, by ( ∆ c , κ ǫ ) : (cid:3) ( ∆ ⊗ C , id ⊗ κ ǫ ) ( C l n , κ ǫ ) : (cid:3) ( Cl n ⊗ C , id ⊗ κ ǫ ) . It is important to note that the complexification ∆ ⊗ C is an irre-ducible module for C l k . This is a non-trivial fact that depends on therepresentation theory of Clifford algberas.The orientifold spinor bundles can now be defined by applying thesemi-equivariant associated bundle construction to a semi-equivariant HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 23 principal bundle coming from a ( Spin c , κ ǫ ) -structure and a complexClifford module equipped with an orientifold action. Definition 45.
Let P → Fr ( V ) be an orientifold-Spin c -structure, anddefine the following orientifold bundles:The orientifold spinor bundle /C : (cid:3) P × ( Spin c ( n ) ,κ ǫ ) ( C l n , κ ǫ ) , The reduced orientifold spinor bundle / S : (cid:3) P × ( Spin c ( n ) ,κ ǫ ) ( ∆ c , κ ǫ ) . Note that if one disregards the orientifold action, then an orientifoldspinor bundle is a complex spinor bundle in the usual sense. In thecase of the reduced orientifold spinor bundle, ∆ c is an irreducible mod-ule for C l k , as mentioned above. This implies that, disregarding theorientifold action, the reduced orientifold spinor bundle is a reducedcomplex spinor bundle. Example 46 (The canonical reduced orientifold spinor bundle overa point) . Using Lemma 43 it is possible to construct a ( Spin c , κ ǫ ) -structure P → Fr ( V ) , for the adjoint representation V of ( Z , id ) ⋉ κ ǫ Spin c ( n ) . If dim ( V ) (cid:3) k , then the irreducible Cl n -module ∆ can beused to construct a canonical reduced spinor bundle / S → pt over thepoint orientifold. Example 47 (The canonical reduced orientifold spinor bundle over S k ) . By Lemma 44, each sphere S n has a canonical ( Z , id ) ⋉ κ ǫ Spin c ( n ) -equivariant ( Spin c , κ ǫ ) -structure. If dim ( V ) (cid:3) k , then the irreducibleCl n -module ∆ can be used to construct a canonical reduced spinorbundle / S → S k over the 8-dimensional sphere. This construction isan adaptation, to the orientifold setting, of the Real equivariant spinorbundle defined on S k by Atiyah [2, p. 128].The space of sections of the orientifold spinor bundle carries anaction by sections of an orientifold Clifford bundle C l ( V ) called Cliffordmultiplication . When a ( Spin c , κ ǫ ) -structure P → Fr ( V ) exists, the ori-entifold Clifford bundle can be expressed as an associated bundle C l ( V ) : (cid:3) P × A d c ( Spin c ( n ) ,κ ǫ ) ( C l n , κ ǫ ) of P , and this characterisation can be used to define Clifford multiplica-tion on sections of the associated spinor bundle. Clifford multiplicationon sections is defined in terms of the action of C l n on the model fibre.In order for Clifford multiplication on sections to be well-defined, thisfibrewise definition of Clifford multiplication must be compatible withthe global topology of the base space. In the orientifold setting, Clifford multiplication is also required to be compatible with an orientifold ac-tion on the spinor bundle, and a canonical orientifold action on C l ( V ) .The ( Spin c , κ ǫ ) -structure used to construct an orientifold spinor bundleensures that both of these requirements are fulfilled. Thus, the benefitof working on semi-equivariance and ( Spin c , κ ǫ ) -orientiation is finallyobserved. In what follows, consider sections of associated bundles tobe represented by equivariant maps from the principal bundle P of anunderlying ( Spin c , κ ǫ ) -structure P → Fr ( V ) into the semi-equivariantfibre, as in Lemma 60. Proposition 48.
Sections ϕ ∈ Γ ( C l ( V )) of the orientifold Clifford bundle actfrom the left on the sections ψ ∈ Γ ( /C) of the orientifold spinor bundle by ( ϕψ )( p ) (cid:3) ϕ ( p ) ψ ( p ) . This action is well-defined and satisfies γ ( ϕψ ) (cid:3) ( γϕ )( γψ ) .Proof. Multiplication is well-defined, as ( ϕψ )( p g ) (cid:3) ϕ ( p g ) ψ ( p g ) (cid:3) ( g − ϕ ( p ) g )( g − ψ ( p )) (cid:3) g − ϕ ( p ) ψ ( p ) (cid:3) g − ( ϕψ )( p ) . Compatibility with the orientifold actions is verified using Lemma 60, ( γ ( ϕψ ))( p ) (cid:3) γ ( ϕψ )( γ − p ) (cid:3) γ ( ϕ ( γ − p ) ψ ( γ − p )) (cid:3) ( γϕ ( γ − p ))( γψ ( γ − p )) (cid:3) ( γϕ )( p )( γψ )( p ) (cid:3) (( γϕ )( γψ ))( p ) . Sections of the orientifold Clifford bundle act on sections of the re-duced orientifold spinor bundle in the same way. One can also checkthat the Clifford multiplication between sections of the orientifold Clif-ford bundle is well-defined and compatible with the orientifold action.Because the orientifold spinor bundle has ( C l n , κ ǫ ) as its modelfibre, it carries a right action by elements of C l n . This right action issometimes described as a multigrading [15, pp. 379-380]. Proposition 49.
An element ϕ ∈ C l n acts from the right on sections ψ ∈ Γ ( /C) by ( ψϕ )( p ) (cid:3) ψ ( p ) ϕ. For γ ∈ Γ , this action satisfies γ ( ψϕ ) (cid:3) ( γψ )( γϕ ) . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 25
Proof.
Consider ϕ as a constant section of the trivial orientifold bundle P × id ( G ,θ ) ( C l n , κ ǫ ) . The right action is well-defined, ( ψϕ )( p g ) (cid:3) ψ ( p g ) ϕ ( p g ) (cid:3) g − ψ ( p ) ϕ ( p ) (cid:3) g − ( ψϕ )( p ) . It is also compatible with the orientifold actions, ( γ ( ψϕ ))( p ) (cid:3) γ ( ψϕ )( γ − p ) (cid:3) γ ( ψ ( γ − p ) ϕ ( γ − p )) (cid:3) ( γψ ( γ − p ))( γϕ ( γ − p )) (cid:3) ( γψ )( p )( γϕ )( p ) (cid:3) (( γψ )( γϕ ))( p ) . Similar considerations show that there is also a right action of C l n on C l ( V ) which is compatible with their orientifold actions.3.2. Connections in Orientifold Spinor Bundles.
In order to definean orientifold Dirac operator, a semi-equivariant connection 1-form isneeded for the semi-equivariant principal ( Spin c ( n ) , κ ǫ ) -bundle P ofthe ( Spin c , κ ǫ ) -structure P → Q underlying the orientifold spinor bun-dle. Such a form can be obtained by using Proposition 39 to extend thelifting ϕ : P → Q to a lifting P → Q × X L , where L is a semi-equivariantprincipal ( U ( ) , κ ǫ ) -bundle. A semi-equivariant connection form canthen be constructed on Q × X L , using the averaging process of Proposi-tion 62, and lifted to P , using the relationship between the Lie algebras spin c ( n ) and so ( n ) ⊕ u ( ) . In the next proposition, q denotes the squaremap of Diagram (2). Proposition 50.
The map ( Ad c × q ) ∗ : spin c ( n ) (cid:3) spin ( n ) ⊕ u ( ) → so ( n ) ⊕ u ( ) is an isomorphism, and satisfies ( Ad c × q ) ∗ ◦ ( id × κ ǫ ) ∗ (cid:3) ( id × κ ǫ ) ∗ ◦ ( Ad c × q ) ∗ . Proof.
That ( Ad c × q ) ∗ is an isomorphism is a standard result [10, p. 18-20,29]. The isomorphism can be written down explicitly by making thefollowing identifications(a) so ( n ) can be identified with the real n × n skew-symmetric ma-tricies. A basis for the skew-symmetric matricies is defined by (cid:8) E ij | ≤ i < j ≤ n (cid:9) where E ij is the n × n matrix with all entriesequal to 0 except for the ( i , j ) th and ( j , i ) th entry, which are equalto 1 and − (b) spin ( n ) can be identified with the linear subspace Λ ⊂ Cl n spanned by the elements (cid:8) e i e j | ≤ i < j ≤ n (cid:9) , see [10, p. 18].(c) u ( ) can be identified with R .With these identifications, ( Ad c × q ) ∗ is the map ( Ad c × q ) ∗ : spin ( n ) ⊕ u ( ) → so ( n ) ⊕ u ( )( e i e j , t ) 7→ ( E ij , t ) , see [10, pp. 19-20,29]. Also, the Γ -actions on spin ( n ) ⊕ u ( ) and so ( n ) ⊕ u ( ) are ( id ⊕ κ ǫ ) ∗ : spin ( n ) ⊕ u ( ) → spin ( n ) ⊕ u ( )( e i e j , t ) 7→ ( e i e j , ι ǫ ( t ))( id ⊕ κ ǫ ) ∗ : so ( n ) ⊕ u ( ) → so ( n ) ⊕ u ( )( E ij , t ) 7→ ( E ij , ι ǫ ( t )) , where ι ǫ : R → R is the involutive action induced by ι : t
7→ − t ∈ R .Examining these maps, it is clear that ( Ad c × q ) ∗ ◦ ( id × κ ǫ ) ∗ (cid:3) ( id × κ ǫ ) ∗ ◦ ( Ad c × q ) ∗ . Proposition 51.
Let ϕ Q : P → Q be a ( Spin c , κ ǫ ) -structure. The semi-equivariant principal bundle P carries a Γ -semi-equivariant connection -form.Proof. By Proposition 39, there exists a lifting ϕ Q × ϕ L : P → Q × X L by Ad c × q , where L is a semi-equivariant principal ( U ( ) , κ ǫ ) -bundle.The equivariant principal bundle Q has an equivariant connection 1-form ω Q : TQ → so ( n ) determined by an equivariant metric. Thesemi-equivariant principal bundle L has a semi-equivariant connection1-form ω L : TL → u ( ) constructed by applying Proposition 62 to anychoice of connection 1-form for L . Together, these two connection1-forms define a semi-equivariant connection 1-form ω Q ⊕ ω L : T ( Q × X L ) → so ( n ) ⊕ u ( ) . Using the ( Spin c , κ ǫ ) -structure ϕ and Proposition 50, the connection1-form ω Q ⊕ ω L can be lifted to a connection 1-form ω : TP → spin c ( n ) v
7→ ( Ad c × q ) − ∗ ◦ ( ω Q ⊕ ω L ) ◦ ( ϕ Q × ϕ L ) ∗ ( v ) . The semi-equivariance of ω follows from the semi-equivariance of ω Q ⊕ ω L , and the equivariance of ( ϕ Q × ϕ L ) ∗ and ( Ad c × q ) ∗ . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 27
The next proposition shows that the connection 1-form constructedby Proposition 51 defines a covariant derivative on the orientifoldspinor bundle that is equivariant with respect to the action of Γ . Inthis proposition, sections will be considered as maps ψ : P → C l n satisfying ψ ( gp ) (cid:3) g − ψ ( p ) , and will be acted on by the Γ -action de-fined in Lemma 60. From the point of view of the exterior covariantderivative, these maps are order-zero tensorial forms ψ ∈ Λ ( P , C l n ) .For the details of tensorial forms and exterior covariant derivatives, see[10, §B.3-4] [22, §II.5]. Proposition 52.
Let ϕ : P → Q be a ( Spin c , κ ǫ ) -structure. The semi-equivariant connection -form ω , defined on P by Proposition 51, determinesan exterior covariant derivative d ω : Λ ( P , C l n ) → Λ ( P , C l n ) that satisfies the condition d ω ( κ ǫ ( γ ) ◦ ψ ◦ η γ − ) (cid:3) κ ǫ ( γ ) ◦ d ω ψ ◦ ( η γ − ) ∗ , where ψ ∈ Λ ( P , C l n ) , η is the Γ -action on P , and κ ǫ is the conjugation actionon C l n .Proof. The vertical projection associated to the connection form ω isdefined by π V | p : (cid:3) ( R p ) ∗ ◦ ω : TP p → TP p . Therefore, the exterior covariant derivative can be written as d ω ψ ( v ) (cid:3) d ψ ◦ π H ( v ) (cid:3) d ψ ( v ) − d ψ ◦ π V ( v ) (cid:3) d ψ ( v ) − d ψ ◦ ( R p ) ∗ ◦ ω ( v ) , (5)where v ∈ TP p , ψ ∈ Λ ( P , C l n ) , and π H is the horizontal projection.The first term of the decomposition (5) is equivariant, as the propertiesof the exterior derivative imply that d ( κ ǫ ( γ ) ◦ ψ ◦ η γ − ) (cid:3) κ ǫ ( γ ) ◦ d ψ ◦ ( η γ − ) ∗ . The semi-equivariance of P implies the identity ( η γ ) ∗ ◦ ( R p ) ∗ (cid:3) ( R γ p ) ∗ ◦( θ γ ) ∗ . Together with the the semi-equivariance of ω , this implies that d ( κ ǫ ( γ ) ◦ ψ ◦ η γ − ) ◦ ( R p ) ∗ ◦ ω (cid:3) κ ǫ ( γ ) ◦ d ψ ◦ ( η γ − ) ∗ ◦ ( R p ) ∗ ◦ ω (cid:3) κ ǫ ( γ ) ◦ d ψ ◦ ( R γ − p ) ∗ ◦ ( θ γ − ) ∗ ◦ ω (cid:3) κ ǫ ( γ ) ◦ d ψ ◦ ( R γ − p ) ∗ ◦ ω ◦ ( η γ − ) ∗ . Therefore, the second term of the decomposition (5) is also equivariant.
Proposition 52 applies equally well to the reduced orientifold spinorbundle if C l n is replaced with ∆ c .As in the non-equivariant case, the exterior covariant derivative isalso equivariant with respect to the right action of C l n on the orientifoldspinor bundle. Proposition 53.
Let ϕ : P → Q be a ( Spin c , κ ǫ ) -structure. The semi-equivariant connection -form ω , defined on P by Proposition 51, determinesan exterior covariant derivative d ω : Λ ( P , C l n ) → Λ ( P , C l n ) that satisfies d ω ( ψϕ ) (cid:3) d ω ( ψ ) ϕ, for ψ ∈ Λ ( P , C l n ) and ϕ ∈ C l n . Dirac Operators on Orientifolds.
At this stage, all of the prelimi-nary constructions have been completed. It is now possible to constructthe orientifold Dirac operator and reduced orientifold Dirac operator.
Definition 54.
Let ∇ L denote the connection associated to a ( Spin c , κ ǫ ) -structure P → Fr ( T M ) by Proposition 51, and µ denote Clifford mul-tiplication by sections of T ∗ M ≃ T M ⊂ C l ( T M ) . Define the orientifoldDirac operator and reduced orientifold Dirac operator , respectively, by /D : (cid:3) µ ◦ ∇ L : Γ ( /C) → Γ ( T ∗ M ⊗ /C) → Γ ( /C) , / D : (cid:3) µ ◦ ∇ L : Γ (/ S ) → Γ ( T ∗ M ⊗ / S ) → Γ (/ S ) . The orientifold Dirac operator and reduced orientifold Dirac oper-ator are complex Dirac operators, in the usual sense. However, theyare equivariant with respect to the orientifold actions on their spinorbundles. Thus, when ǫ : Γ → Z is non-trivial, they have anti-linearsymmetries. Proposition 55.
The orientifold Dirac operator is equivariant with respect tothe left action of Γ on sections of /C , /D( γψ ) (cid:3) γ /D( ψ ) . Proof.
This follows from Propositions 48 and 52.The same arguments show that the reduced orientifold spinor bun-dle is also Γ -equivariant. In addition to Γ -equivariance, the orientifoldDirac operator is equivariant with respect to the right action of ( C l n , κ ǫ ) on the orientifold spinor bundle. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 29
Proposition 56.
The orientifold Dirac operator is equivariant with respect tothe right action of C l n on sections of /C , /D( ψϕ ) (cid:3) /D( ψ ) ϕ. Proof.
This follows from Propositions 49 and 53.Note, in particular, that left and right equivariance together implythat the index of /D consists of vector spaces which are both Cliffordmodules and orientifold representations of ( Γ , ǫ ) .The main aim of this paper is now complete, and the followingtheorem has been proved. Theorem 57.
Let X be an orientifold with orientifold group ( Γ , ǫ ) .(a) If W ( Γ ,ǫ ) ( X ) (cid:3) , then X carries an orientifold Dirac operator.(b) If W ( Γ ,ǫ ) ( X ) (cid:3) and dim ( X ) (cid:3) , then X carries a reduced orientifoldDirac operator.In particular, if X is an k -dimensional Real manifold and W ( Z , id ) ( X ) (cid:3) ,then X carries a reduced Real Dirac operator.
4. Related Work and
Applications
To put the construction of the orientifold Dirac operator in context, it isworth breifly recalling the position that Spin-structures and Spin-Diracoperators occupy in the K -theory of real vector bundles. The central-ity of Spin-structures and the Spin-Dirac operator in the K -theory ofreal vector bundles stems from their role in Atiyah’s index theoreticproof of the Thom isomorphism theorem [2], which provides an iso-morphism KO ( X ) → KO ( V ) between the KO -theory of a manifoldand any 8 k -dimensional Spin-oriented real vector bundle V → X .The proof proceeds by compactifying the fibres of V into a family of8 k -dimensional spheres, each of which is equipped with a canonicalreduced Dirac operator. The families index map KO ( V ) → KO ( X ) associated to this family of Dirac operators is then shown to providean inverse to the Thom map. This method of proof naturally accomo-dates additional symmetries, provided that the appropriate analogueof Spin-structure and Dirac operator can be determined. In this way,Atiyah was able to prove the equivariant Thom isomorphism [2]. The ( Spin c , κ ǫ ) -structures and orientifold Dirac operators defined in thispaper play an exactly analogous role in the K -theory of orientifoldbundles. These allow Atiyah’s argument to be extended, providing a proof of the corresponding Thom isomorphism theorem for orien-tifold K -theory [20, Theorem 4.30]. They also provide a basis for thedefinition of geometric orientifold K -homology [20, Chapter 6].Orientation conditions in K -theory, such as ( Spin c , κ ǫ ) -orientibility,are closely related to the topic of twisted K -theory. The twisted K -theory of Real topological groupoids has been studied by Moutuouusing a Čech cohomology for Real groupoids [25]. In a more alge-braic context, Karoubi and Weibel have studied an equivariant twisted K -theory that includes KR -theory as a special case [18]. Another ap-proach by Hekmati et al. [14], motivated by applications to orientifoldstring theories, studies KR -orientiation and twisting using a Real sheafcohomology theory. The work of Freed and Moore on topological insu-lators [9] also treats twistings of K -theory in the presence of symmetry.Orientifold string theory and the classification of topological insu-lators are two areas in which the constructions of this paper have poten-tial applications. The connection between the present investigation andstring theory begins with the classification of D-brane charges using K -theory, as described in [24, 30]. Results in index theory allow one topass from K -theory to an analytic K -homology theory in which classesare represented by elliptic operators. Each class in this K -homologytheory may be represented by a Dirac operator that has been twistedwith a vector bundle. By replacing these Dirac operators with classesformed from the Spin c -structures and vector bundles used to constructthem, it is possible to define a K -homology theory in entirely geometricterms [5, 4, 6]. This characterisation of D-brane charge is of interest, asthe geometric data comprising such a K -homology class has physicalinterpretations [3] [28, §4]. Three types of orientifold string theoriesare listed in [30, p. 26-27], along with the corresponding K -theories thatclassifying the associated D-brane charges. In the first of these, D-branecharges are classified by KR -theory. The ( Spin c , κ ǫ ) -structure and ori-entifold Dirac operators constructed in this paper provide the ingre-dients neccesary to generalise the above discussion to KR -theory andthe K -theory of orientifold bundles. The two other possibilities listedin [30, p. 26-27] involve K -theory with sign-choice. This K -theory hasbeen studied by Doran et al. [8] using methods from non-commutativegeometry. The K -theory with sign-choice is a subgroup of the K -theoryorientifold bundles. Many of the constructions discussed in the presentpaper could be modified to incorporate sign-choice structures.In recent years, there there has been much interest in the classi-fication of topological insulators. These classification attempts leadnaturally to the consideration of topological invariants which respectanti-linear symmetries [17, 11, 12, 13, 26]. Contact with Clifford al- HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 31 gebras and K -theory has been made through the work of Kitaev [19].Another framework for studying topological insulators, using twisted K -theories, has been described by Freed and Moore [9]. The K -theoryof orientifold bundles is a primary example within their framework.Thus, it appears that there is potential for index invariants derivedfrom the orientifold Dirac operator to be applied to the classification oftopological insulators. A. Semi-equivariant Constructions
A.1.
Semi-equivariance and Associated Bundles.
The constructionof associated bundles from semi-equivairant principal bundles differsslightly from the corresponding equivariant construction. When form-ing an equivariant vector bundle from an equivariant principal bun-dle, the only requirement on the model fibre is that it carries carriesan action of the structure group G . However, when forming a vectorbundle from a semi-equivariant principal bundle, it is neccesary to usea model fibre that carries both an action of the structure group G andan action of the equivariance group Γ . As on the semi-equivariantprincipal bundle, these two actions are required to combine into anaction of the semi-direct product group Γ ⋉ θ G . Although the actionof the equivariance group G on the model fibre is required to be linear,the action of the equivariance group Γ is not. This makes it possibleto construct associated bundles with Γ -actions that are not linear. Inparticular, it is possible to construct complex vector bundles equippedwith linear/anti-linear actions as semi-equivariant associated bundles. Definition 58.
Let P be a Γ -semi-equivariant principal ( G , θ ) -bundle.A semi-equivariant fibre for P is a vector space V equipped with a linearaction of G and an action of Γ by diffeomorphisms, such that γ ( gv ) (cid:3) ( γ g )( γ v ) . Definition 59.
Let P be a Γ -semi-equivariant principal ( G , θ ) -bundle,and V be a semi-equivariant fibre for P . The semi-equivariant associatedbundle is the vector bundle P × ( G ,θ ) V : (cid:3) P × V /∼ where ( p , v ) ∼ ( p g − , gv ) . This bundle carries an action of Γ definedby γ ( p , v ) : (cid:3) ( γ p , γ v ) . Note that the Γ -action on P × ( G ,θ ) V is well-defined because γ [ p g − , gv ] (cid:3) [ γ ( p g − ) , γ ( gv )] (cid:3) [( γ p )( γ g ) − , ( γ g )( γ v )] (cid:3) [ γ p , γ v ] (cid:3) γ [ p , v ] . Sections of associated bundles are often represented as equivariantmaps from the principal bundle into the model fibre. It is sometimesuseful to express the action of Γ on a section in this way. HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 33
Lemma 60.
Sections of P × ( G ,θ ) V are in bijective correspondence with maps ψ : P → V such that ψ ( p g ) (cid:3) g − ψ ( p ) . The Γ -action on sections of P × ( G ,θ ) V corresponds to the Γ -action ( γψ )( p ) (cid:3) γψ ( γ − p ) on these maps.Proof. A map ψ : P → V with ψ ( p g ) (cid:3) g − ψ ( p ) corresponds to thesection of P × ( G ,θ ) V defined by s ( p ) : (cid:3) [ p , ψ ( p )] . The Γ -action on sucha section is ( γ s )( p ) : (cid:3) γ s ( γ − p ) (cid:3) γ [ γ − p , ψ ( γ − p )] (cid:3) [ γγ − p , γψ ( γ − p )] (cid:3) [ p , γψ ( γ − p )] . Thus, the corresponding map on P is p γψ ( γ − p ) .A.2. Semi-equivariant Connections.
In the smooth non-equivariantsetting, a connection for a principal G -bundle P can be expressed asa g -valued 1-form on the tangent space TP , where g is the Lie algebraof the structure group G [22, Chapter 2], [10, Appendix B]. A Γ -semi-equivariant ( G , θ ) -principal bundle [21, §2] has a Γ -group ( G , θ ) asits structure group. The differentials ( θ γ ) ∗ of the Γ -action on G forma Γ -action on the Lie algebra g . A connection in a semi-equivariantprincipal bundle must be compatible with this action if it is to producean equivariant connection in an associated bundle. The definition ofa semi-equivariant connection 1-form is given below, along with anaveraging proceedure that can be used to construct semi-equivariantconnections. In what follows, let R g ( p ) (cid:3) R p ( g ) : (cid:3) p g denote themultiplication maps associated to the right action on a principal G -bundle P . Also, let R g ( h ) : (cid:3) h g denote the right action of G on itself.Note that ( R p ) ∗ ( A e ) defines the vector field induced on P by an element A ∈ g , and the adjoint map on g may be expressed as Ad g − (cid:3) ( R g ) ∗ . Definition 61.
Let ( P , η ) be a smooth Γ -semi-equivariant principal ( G , θ ) -bundle with Γ -action η , and let g be the Lie alegebra of G . A Γ -semi-equivariant connection -form on P is a Lie algebra valued 1-form ω : TP → g such that for all γ ∈ Γ , g ∈ G , A ∈ g , and p ∈ P , ω ◦ ( R p ) ∗ ( A e ) (cid:3) A ω ◦ ( R g ) ∗ (cid:3) ( R g ) ∗ ◦ ω ω ◦ ( η γ ) ∗ (cid:3) ( θ γ ) ∗ ◦ ω. When Γ is finite, a semi-equivariant connection can be constructedfrom a given connection by a twisted averaging procedure. Proposition 62.
Let Γ be a finite Lie group, and suppose that P is a smooth Γ -semi-equivariant principal ( G , θ ) -bundle with Γ -action η . If ω : TP → g is a connection form on P , then ω Γ : (cid:3) Õ γ ∈ Γ ( θ γ ) ∗ ◦ ω ◦ ( η γ − ) ∗ is a Γ -semi-equivariant connection on P .Proof. First note that, as θ is an automorphism and P is semi-equivariant,identities are induced between the differentials of the various actions.For γ ∈ Γ , g , h ∈ G , and p ∈ P γ ( h g ) (cid:3) ( γ h )( γ g ) (cid:3) ⇒ ( θ γ ) ∗ ◦ ( R g ) ∗ (cid:3) ( R γ g ) ∗ ◦ ( θ γ ) ∗ γ ( p g ) (cid:3) ( γ p )( γ g ) (cid:3) ⇒ ( ( η γ ) ∗ ◦ ( R g ) ∗ (cid:3) ( R γ g ) ∗ ◦ ( η γ ) ∗ ( η γ ) ∗ ◦ ( R p ) ∗ (cid:3) ( R γ p ) ∗ ◦ ( θ γ ) ∗ . To check that ω Γ is a connection, first observe that the condition ω Γ ◦( R p ) ∗ ( A e ) (cid:3) A holds, ( θ γ ) ∗ ◦ ω ◦ ( η γ − ) ∗ ◦ ( R p ) ∗ ( A e ) (cid:3) ( θ γ ) ∗ ◦ ω ◦ ( R γ − p ) ∗ ◦ ( θ γ − ) ∗ ( A e ) (cid:3) ( θ γ ) ◦ ω ◦ ( R γ − p ) ∗ (( θ γ − ) ∗ ( A ) e ) (cid:3) ( θ γ ) ∗ ◦ ( θ γ − ) ∗ ( A ) (cid:3) A . The condition ω Γ ◦ ( R g ) ∗ (cid:3) ( R g ) ∗ ◦ ω Γ also holds, as ( θ γ ) ∗ ◦ ω ◦ ( η γ − ) ∗ ◦ ( R g ) ∗ (cid:3) ( θ γ ) ∗ ◦ ω ◦ ( R γ − g ) ∗ ◦ ( η γ − ) ∗ (cid:3) ( θ γ ) ∗ ◦ ( R γ − g ) ∗ ◦ ω ◦ ( η γ − ) ∗ (cid:3) ( R g ) ∗ ◦ ( θ γ ) ∗ ◦ ω ◦ ( η γ − ) ∗ . Finally, semi-equivariance holds, as ω Γ ◦ ( η γ ) ∗ (cid:3) ( Õ γ ∈ Γ ( θ γ ) ∗ ◦ ω ◦ ( η γ − ) ∗ ) ◦ ( η γ ) ∗ (cid:3) Õ γ ∈ Γ ( θ γ ) ∗ ◦ ω ◦ ( η γ − γ ) ∗ (cid:3) Õ γ ∈ Γ ( θ γγ − ) ∗ ◦ ω ◦ ( η γ ) ∗ (cid:3) ( θ γ ) ∗ ◦ ( Õ γ ∈ Γ ( θ γ − ) ∗ ◦ ω ◦ ( η γ ) ∗ ) (cid:3) ( θ γ ) ∗ ◦ ω Γ . HE CONSTRUCTION OF DIRAC OPERATORS ON ORIENTIFOLDS 35
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