Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
aa r X i v : . [ h e p - t h ] N ov YITP - 07 - 53
HOMOLOGY AND K–THEORY METHODS FOR CLASSESOF BRANES WRAPPING NONTRIVIAL CYCLES
A. A. Bytsenko ( a ) ( b ) 1( a ) Departamento de F´ısica, Universidade Estadual de LondrinaCaixa Postal 6001, Londrina-Paran´a, Brazil ( b ) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract
We apply some methods of homology and K-theory to special classes of branes wrapping ho-mologically nontrivial cycles. We treat the classification of four-geometries in terms of compactstabilizers (by analogy with Thurston’s classification of three-geometries) and derive the K-amenability of Lie groups associated with locally symmetric spaces listed in this case. Morecomplicated examples of T-duality and topology change from fluxes are also considered. Weanalyse D-branes and fluxes in type II string theory on C P × Σ g × T with torsion H − flux anddemonstrate in details the conjectured T-duality to R P × X with no flux. In the simple caseof X = T , T-dualizing the circles reduces to duality between C P × T × T with H − flux and R P × T with no flux.Keywords: branes; homological and K-theory methods [email protected] ontents C ∗ − algebras . . . . . . . . . . . . . . . . . . . . . . . 114.2 The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 The four-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 R P × T . . . . . . . . . . . . . . . . . . . . . 185.2 T-dualizing on C P × Σ g × T . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Introduction
The problem of classifying geometries is one of the main problem in complex analysis and inmathematics as a whole, and plays a fundamental role in physical models. Every one-dimensionalmanifold is either S (closed, i.e. compact with empty boundary) or R (open), with a uniquetopological, piecewise linear and smooth structure and orientation. All curves of genus zerocan be uniformized by rational functions, all those of genus one can be uniformized by ellipticfunctions, and all those of genus more than one can be uniformized by meromorphic functions,defined on proper open subsets of C . This result, due to Klein, Poincar´e and Koebe, is one ofthe deepest achievements in mathematics. A complete solution of the uniformization problemhas not yet been obtained, excepted for the one-dimensional complex case. However, there wereadvances in this problem, which have been essential to understand the foundations of topologicalmethods, covering spaces, existence theorems for partial differential equations, existence anddistorsion theorems for conformal mappings, etc.In accordance with Klein-Poincar´e uniformization theorem, each Riemann surface can berepresented (within a conformal equivalence) in the form Σ / Γ, where Σ is one of the threecanonical regions, namely: The extended plane ¯ C (the sphere S ), the plane C ( R ), or the disk,and Γ is a discrete group of M¨obius automorphisms of Σ acting freely there. Riemann surfaceswith such coverings are elliptic, parabolic and hyperbolic type, respectively. This theorem ad-mits a generalization also to surfaces with branching. With the help of standard uniformizationtheorems and decomposition theorems [1], one can construct and describe all the uniformizationsof Riemann surfaces by Kleinian groups. Furthermore, by using the quasiconformal mappings,one can obtain an uniformization theorem of more general character . Namely, it is possibleto prove that several surfaces can be uniformized simultaneously. For any closed orientabletwo-dimensional manifold Σ Γ the following result holds: Every conformal structure on Σ Γ isrepresented by a constant curvature geometry. The only simply connected manifolds with con-stant curvature are S or R or H and Σ Γ can be represented as Σ / Γ, where Γ is a group ofisometries.An important progress in the three-dimensional case has been made by Thurston [2]. Toanalyse this case we should consider the eight geometric structure in the classification of three-manifolds introduced by Thurston. We also note the recent asserts for Ricci flow on class ofthree-manifolds. The Ricci flow with surgery was considered in [3]. A canonical Ricci flowdefined on largest possible subset of spacetime has been constructed in [4, 5].In this paper we will analyse in detail the K-theory groups which are the more appropriatedarena to classify the D-branes wrapping topologically nontrivial manifolds. We contemplate var-ious three- and four-geometries giving the relevant meaning of low dimensional brane cycles anddescribing some new results on K-amenability in the list of four-geometries. We concentrate ouranalysis on application of homological and K-theory methods to branes and Ramond-Ramond(RR) flux in type II string theory. The RR fields are typically supported on D-branes and takevalues in appropriate K-theory groups [6]. These facts have prompted intensive investigationsin both the mathematical and physical literature into the properties and definitions of variousK-theory groups.This paper is outlined as follows. The case of two-geometries is shortly discussed in Section2.1 and then in Section 2.2 we present the Thurston’s list of three-geometries. This list has beenorganized in terms of the compact stabilizers Γ x of x ∈ X isomorphic to SO (3), SO (2) or trivialgroup { } . The analogue list for four-geometries and the corresponding stabilizer-subgroups of This fact is related to Techm¨uller spaces. SO (4), U (2), SO (2) × SO (2), SO (3), SO (2), S , and { } are analysed in Section2.3.The required mathematical tools considered in the paper are exposed in the Section 3. Wealso refer the reader to the Appendix where the necessary material on Eilenberg-MacLane spacesis presented. The Kasparov’s KK-pairings and the concept of K-amenable groups are consideredin the Section 3.1 and the Section 3.2, respectively.The different aspects of K-group theory are formulated in the Section 4. Twisted crossedproduct of C ∗ − algebras and twisted K-groups are considered in the Section 4.1.Remind that some general theorems about the K-groups of C ∗ − algebras have been ob-tained in [7, 8, 9], while perfect analysis of Baum-Connes type conjectures on the K-theoriesof twisted C ∗ − algebras was considered in [10, 11]. In the special case of general statements, K ∗ ( C ∗ ( Z n , σ )) ∼ = K ∗ ( C ∗ ( Z n )) ∼ = K ∗ ( T n ) for any multiplier (i.e. group two-cocycle) σ on Z n .The twisted group C ∗ − algebras C ∗ ( Z n , σ ) have been called noncommutative tori . In [12, 13] thiscalculation was generalized for K-groups of the twisted group C ∗ − algebras of uniform latticesin solvable groups. Namely, if Γ is a uniform lattice in a solvable Lie group G , then K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ +dim G (Γ \ G, δ ( B σ )) , where σ is any multiplier on Γ, K ∗ (Γ \ G, δ ( B σ )) denotes the twisted K-theory (see for detail [14])of a continuous trace C ∗ − algebra B σ with spectrum Γ \ G , and δ ( B σ ) ∈ H (Γ \ G, Z ) denotes theDixmier-Douady invariant of B σ . This result has been prooved by using the Packer-Raeburnstabilization trick [12] and the Thom isomorphism theorem for the K-theory of C ∗ algebras.Then in [15] the main theorem of [12, 13] has been extended to the case when Γ is a lattice ina K-amenable Lie group G [16]. For such G and Γ, K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ ( C ∗ r (Γ , σ )) , K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ +dim( G/K ) (Γ \ G/K, δ ( B σ )) , where K is a maximal compact subgroup of G , σ is any multiplier on Γ, K ∗ (Γ \ G/K, δ ( B σ )) isthe twisted K-theory of a continuous trace C ∗ − algebra B σ with spectrum Γ \ G/K .For the three-dimensional case these results are described in the Section 4.2. Then, in theSection 4.3 we proof the K-amenability for a set of Lie groups associated with locally symmetricspaces listed in the four-dimensional case. We formulate the following statement in the form ofconjecture: If K (Γ ,
1) is a connected, compact, four-dimensional manifold which is an Eilenberg-MacLane space with fundamental group Γ, then for any multiplier σ ∈ H (Γ , U (1)) on Γ K j ( C ∗ r (Γ , σ )) ∼ = K j ( C ∗ r (Γ)) ∼ = K j +1 ( K (Γ , , j = 0 , . The properties of the T-duality for principal higher rank torus bundles with H-flux havebeen studied in [17]. This construction of duality can be argued from the so-called generalizedGysin sequence. T-duality for torus bundles comes with the isomorphism of twisted cohomologyand twisted cyclic homology, it was analysed via noncommutative topology in [18, 19]. It hasbeen argued [20, 21] that for type II string theory a general principal torus bundle with generalH-flux is a bundle of noncommutative, nonassociative tori. Useful theorems to perform theT-duality computations for circle bundles with H-fluxes are given in [22, 23]. In the Section 5we discuss some more complicated examples of T-duality for circle bundles and topology changefrom fluxes, taking into account methods of homology and K-theory. We apply to D-branesclassification and fluxes in type II string theory on C P × Σ g × T with torsion H − flux (similarmodel has been considered in [24, 21]) and demonstrate in details the conjectured T-duality to R P × X with no flux. In the simple case of X = T , T-dualizing the circles reduces to dualitybetween C P × T × T with H − flux and R P × T with no flux.4able 2.1. Two-space integral homologiesHomologies n= 0 n= 1 n = 2 H n ( X oriented ) Z Z ⊕ g Z H n ( X non − oriented ) Z Z ⊕ g − ⊕ Z / Two-manifolds are complete classified: their piecewise linear and smooth structures are uniqueand depend only on the homeomorphic type. The homeomorphism type is determined by theirfundamental group. If X is a closed orientable two-manifold, then it is either S , T = S × S or an g − fold connected sum ( T ) g . If two-manifold X is not orientable, then it is the realprojective plane R P or g − fold connected sum thereof, ( R P ) g . Surfaces of g ≥ g are presented in Table 2.1. If X is the orientable surface then X = S for g = 0, and X = T for g = 1. For non-orientablesurface X we have X = R P and X = the Klein bottle for g = 1 and 2, respectively. Fororientable and non-orientable cases the Euler characteristic χ ( X ) = P ℓ ( − ℓ rk H ℓ ( X ) satisfies χ ( X ) = 2 − g and χ ( X ) = 2 − g, respectively. Following the presentation of [2, 25], by a geometry or a geometric structure we mean a pair( X, Γ) that is a manifold X and a group Γ acting transitively on X with compact point stabilizers(following [2] we also propose that the interior of every compact three-manifold has a canonicaldecomposition into pieces which have geometric structure). Two geometries ( X, Γ) and ( X ′ , Γ ′ )are equivalent if there is a diffeomorphism of X with X ′ which throws the action of Γ onto theaction of Γ ′ . In particular, Γ and Γ ′ must be isomorphic. Assume that: • The manifold X is simply connected. Otherwise it will be sufficient to consider a naturalgeometry ( e X, e Γ), e X being the universal covering of X and e Γ denoting the group of alldiffeomorphisms of e X which are lifts of elements of Γ. • The geometry admits a compact quotient. In another words, there exists a subgroup b Γ ofΓ which acts on X as a covering group and has compact quotient. • The group Γ is maximal. Otherwise, if Γ ⊂ Γ ′ then any geometry ( X, Γ) would be thegeometry ( X, Γ ′ ) at the same time. The integer g is called the genus of the surface X and determines it and its fundamental group up tohomeomorphism; we assume that g ≥ X = G/K = ( R (Euclidean space) , S (spherical space) , H (hyperbolic space) , H × R , S × R , ^ SL (2 , R ) , N il , S ol ) Remark 2.1
This conjecture follows from considering the identity component of the isotropygroup Γ x of X through a point x . Γ x is a compact, connected Lie group, and there are threecases: Γ x = SO (3) , SO (2) and { } . R . Γ x = SO (3) . In this case the space X is a space of constant curvature: R , S (modelledon R ) or H (which can be modelled on the half-space R × R + ). R . Γ x = SO (2) . In this case there is a one-dimensional subspace of T x X that invariant under Γ x , which has a complementary plane field P x . If plane field P x is integrable, then X isa product R × S or R × H . If plane field P x is non-integrable, then X is a non-trivialfiber bundle with fiber S : S ֒ → X ։ Σ g ≥ ( ^ SL (2 , R ) − geometry), Σ g stands for a surfaceof genus g , S ֒ → X ։ T ( N il − geometry) or S ֒ → X ։ S ( S − geometry). R . Γ x = { } . In this case we have three-dimensional Lie groups: ^ SL (2 , R ) , N il , and S ol . The first five geometries are familiar objects, so we briefly discuss the last three ones. The group ^ SL (2 , R ) is the universal covering of SL (2 , R ), the three-dimensional Lie group of all 2 × N il is the three-dimensional Lie group of all3 × S ol is the three-dimensional (solvable) group.Many three-manifolds are hyperbolic (according to a famous theorem by Thurston [2]). Forexample, the complement of a knot in S admits an hyperbolic structure unless it is a torus orsatellite knot. Moreover, after the Mostow Rigidity Theorem [26], any geometric invariant of anhyperbolic three-manifold is a topological invariant. If a three-manifold X admits a geometricstructure then the universal cover e X with the induced metric is isometric to one of the eightgeometries above. X can admit more than one geometric structure, but if X is closed and admitsa geometric structure then the geometric structure is unique: e X is isomorphic to one and onlyone of the above geometries (for more detail see [25]). The Euclidean space, being flat, does notlead to a new background in supergravity, i.e. the topological twist is trivial. Our special interestto hyperbolic spaces: It has been shown that for supergravity descriptions of branes wrappingthree-dimensional cycles implies that cycles have to be a constant scalar curvature. The solutionincluding H and its quotients by subgroup of isometry group PSL (2 , C ) ≡ SL (2 , C ) / {± } thereader can found in [27, 28]. The list of Thurston three-geometries has been organized in terms of the compact stabilizers Γ x of x ∈ X isomorphic to SO (3), SO (2) or trivial group { } . The analogue list of four-geometriesalso can be organized (using only connected groups of isometries) as in Table 2.3. Here we havethe four irreducible four-dimensional Riemannian symmetric spaces: sphere S , hyperbolic space H , complex projective space C P and complex hyperbolic space CH (which we may identifywith the open unit ball in C with an appropriate metric). The other cases are more specificand only for the sake of completeness we shall illustrate them.6able 2.3. List of four-geometriesStabilizer-subgroup Γ x Space X SO (4) S , R , H U (2) C P , CH SO (2) × SO (2) S × R , S × S , S × H , H × R , H × H SO (3) S × R , H × R SO (2) N il × R , ] PSL (2 , R ) × R , S ol S F trivial N il , S ol m,n (including S ol × R ) , S ol The nilpotent Lie group N il can be presented as the split extension R ⋊ U R of R by R , where the real 3-dimensional representation U of R has the form U ( t ) = exp( tB ) with B = . In the same way the soluble Lie groups S ol m,n = R ⋊ T m,n R forms on real3-dimensional representations T m,n of R , T m,n ( t ) = exp( tC m,n ), where C m,n = diag( α, β, γ ) and α + β + γ = 0 for α > β > γ . Furthermore e α , e β and e γ are the roots of λ − mλ + nλ − m, n positive integers. If m = n , then β = 0 and S ol m,n = S ol × R . In general, if C m,n ∝ C m ′ ,n ′ ,then S ol m,n ∼ = S ol m ′ ,n ′ . It gives infinity many classes of equivalence. When m n + 18 =4( m + n ) + 27, one has a new geometry, S ol , associated with the group SO (2) of isometriesrotating the first two coordinates. The soluble group S ol , is most conveniently represented as thematrix group b c α a : α, a, b, c ∈ R , α > . Finally, the geometry F is associatedwith the isometry group R ⋊ PSL (2 , R ) and stabilizer SO (2). Here the semidirect productis taken with respect to action of the group PSL (2 , R ) on R . The space F is diffeomorphicto R and has alternating signs of metric. A connection of these geometries with complex andK¨ahlerian structures (preserved by the stabilizer Γ σ ) can be found in [29]. In this section we discuss K-groups of the twisted group C ∗ − algebras, which are relevant tothe definition of a twisted analogue of the Kasparov map, pairing between K-theory and cycliccohomology theory, and which enables us to use K-amenability results for Eilenberg-Maclanespaces. Given a manifold X , let C ( X ) be the commutative C ∗ − algebra (recall that a C ∗ − algebra isa Banach algebra with an involution satisfying the relation || aa ∗ || = || a || ) of all continuouscomplex-valued functions which vanish at infinity on X . The C ∗ − algebra, which categorically7ncodes the topological properties of manifold X , plays a dual role to X in the K-theory of X by the Serre-Swan theorem [30]: e K ℓ ( X ) ∼ = e K ℓ ( C ( X )) , ℓ = 0 , . (3.1)Here e K ℓ ( X ) is the reduced topological K-theory of X : Taking into account that a vector bundleover a point is just a vector space, K (pt) = Z , we can introduce a reduced K-theory in whichthe topological space consisting of a single point has trivial cohomology, e K (pt) = 0, and also e K ( X ) = 0 for any contractible space X . Let us consider the collapsing and inclusion maps: p : X → pt , ι : pt ֒ → X for a fixed base point of X . These maps induce an epimorphism and amonomorphism of the corresponding K-groups: p ∗ : K (pt) = Z → K ( X ) , ι ∗ : K ( X ) → K (pt) = Z . The exact sequences of groups are:0 → Z p ∗ → K ( X ) → e K ( X ) → , → e K ( X ) → K ( X ) ι ∗ → Z . The kernel of the map i ∗ (or the cokernel of the map p ∗ ) is called the reduced K-theory group and is denoted by e K ( X ), e K ( X ) = ker ι ∗ = coker p ∗ . There is a fundamental decomposition K ( X ) = Z ⊕ e K ( X ). When X is not compact, we can define K c ( X ), the K-theory with compactsupport. It is isomorphic to e K ( X ). Let X be a Spin C − manifold, then there is a Poincar´eduality isomorphism [31]: K dim X − ℓ ( X ) ∼ = K cℓ ( X ) , ℓ = 0 , , (3.2)where K cℓ denotes the dual compactly supported K-homology of X .For a finite dimensional manifold X there exists another C ∗ − algebra, which is non-commutativeand can be constructed with the help of the Riemannian metric g . In fact, we can form thecomplex Clifford algebra Cliff( T x X, g x ), where for each x ∈ X the tangent space T x X of X is afinite-dimensional Euclidean space with inner product g x . This algebra has a canonical structureas a finite-dimensional Z − graded C ∗ − algebra. The family of C ∗ − algebras { Cliff( T x X, g x } x ∈ X forms a Z − graded C ∗ − algebra vector bundle Cliff( T x X ) → X , called the Clifford algebra bun-dle of X [32]. Let us define C ( X ) = C ( X, Cliff( T x X )) to be the C ∗ − algebra of continuoussections of the Clifford algebra bundle of X vanishing at infinity. If the manifold X is even-dimensional and has a Spin C − structure then this C ∗ − algebra is Morita equivalent to C ( X ).However, in general, C ( X ) is Morita equivalent to C ( T x X ). Because of the Morita equivalenceof K-theory, it follows that K ℓ ( C ( X )) ∼ = K ℓ ( C ( X )) ∼ = K ℓ ( X ) , ℓ = 0 , . But for odd-dimensionaland spin manifold X this relation is more complicated.Recall that the definition of K-homology involves classifying extensions of the algebra ofcontinuous functions C ( X ) on the manifold X by the algebra of compact operators up to uni-tary equivalence [33]. The set of homotopy classes of operators defines the K-homology group K ( X ), and the duality with K-theory is provided by the natural bilinear pairing ([ E ] , [ D ]) Index D E ∈ Z , where [ E ] ∈ K ( X ) and D E denotes the action of the Fredholm operator D onthe Hilbert space H = L ( U ( X, E )) of square-integrable sections of the vector bundle E → X as D : U ( X, E ) → U ( X, E ). It assumes that the KK-pairing may be the most natural frameworkin this context. The group KK ( A, B ) is a bivariant version of K-theory and it depends on apair of graded algebras A and B . Definition 3.1
Let A and B be C ∗ − algebras. D . A pair ( E , π ) will be called an ( A, B ) − bimodule if E is a Z / Z graded Hilbert B − moduleon which algebra A acting by means of ∗− homomorphism π : A → L ( E ) = End ∗ ( E ) , The reader can find the application of KK-group theory to the classification of branes, for example, in [34, 35]. here ∀ a ∈ A an operator π ( a ) being of degree 0, π ( A ) ⊂ L ( E ) (0) . Let E ( A, B ) be a triple ( E , π, F ) , where ( E , π ) is a A, B − bimodule, F ∈ L ( E ) is a homogeneous operator of degree1, and ∀ a ∈ A : π ( a )( F − ∈ C ( E ) , [ π ( a ) , F ] ∈ C ( E ) , where C ( E ) is the algebra ofcompact operators. D . A triple ( E , π, F ) will be called degenerate if ∀ a ∈ A : π ( a )( F −
1) = 0 , [ π ( a ) , F ] = 0 . D . Let D ( A, B ) be a set of degenerated triples. An element E ( A, B [0 , , where B [0 , is analgebra of continuous functions in B on the interval [0 , , will be called a homotopy in E ( A, B ) . Let us assign a direct sum in E ( A, B ):( E , π, F ) ⊕ ( E ′ , π ′ , F ′ ) = ( E ⊕ E ′ , π ⊕ π ′ , F ⊕ F ′ ) . (3.3)The homotopy classes of E ( A, B ) together with this sum defines the Abelian group KK ( A, B ).It is clear that any degenerate triplet is homotopy equivalent to (0 , , E , π, F ) is equal to ( − E , π, − F ), where − E means that grading on E has to be inverted. f : A → A transfers ( A , B ) − modules into ( A , B ) − modules, and [36] f ∗ : E ( A , B ) → E ( A , B ) , ( E , π, F ) ( E , π ◦ f, F ) , (3.4)On the other hand a ∗− homomorphism g : B → B induces a homomorphism g ∗ : E ( A, B ) → E ( A, B ) , ( E , π, f ) ( E ⊗ g B , π ⊗ , F ⊗ , (3.5)where π ⊗ A → L ( E ⊗ g B ) , ( π ⊗ a )( e ⊗ b ) = π ( a ) e ⊗ b . (3.6) Theorem 3.1
The groups KK ( A, B ) define an homotopy invariant bifunctor from the categoryof separable C ∗ − algebras into the category of Abelian groups. Abelian groups KK ( A, B ) dependcontravariantly on the algebra A and covariantly on the algebra B , in addition KK ( C , B ) = K ( B ) . Definition 3.2
Let A ∈ KK ( A, A ) ( KK ( A, A ) is a ring with unit) denotes the triple class ( A, ι A , where A (1) = A, A (0) = 0 , and ι A : A → C ( A ) ⊂ L ( A ) , ι A ( a ) b = ab, a, b ∈ A . Let usdefine also the map D . τ D : KK ( A, B ) ⊗ KK ( A ⊗ D, B ⊗ D ) . D . τ D (class ( E , π, F )) = class ( E ⊗ D, π ⊗ D , F ⊗ . Theorem 3.2
Kasparov’s pairing, defined by KK ( A, D ) × KK ( D, B ) −→ KK ( A, B ) (3.7) and denoted ( x, y ) x ⊗ D y , satisfies the following properties: T . It depends covariantly on the algebra B and contravariantly on the algebra A . The interesting facts to us are the following relations: KK ∗ ( A := C , B ) = K ∗ ( B ) , KK ∗ ( A, B := C ) = K ∗ ( A ) . . If f : D → E is a ∗− homomorphism, then f ∗ ( x ) ⊗ E y = x ⊗ D f ∗ ( y ) , x ∈ KK ( A, D ) , y ∈ KK ( E, B ) . T . Associative property: ( z ⊗ D y ) ⊗ E z = x ⊗ D ( y ⊗ ) E z , ∀ x ∈ KK ( A, D ) , y ∈ KK ( D, E ) , z ∈ KK ( E, B ) . T . x ⊗ B B = 1 A ⊗ x = x, ∀ x ∈ KK ( A, B ) . T . τ E ( x ⊗ B y ) = τ E ( x ) ⊗ B ⊗ E τ E ( y ) , ∀ x ∈ KK ( A, B ) , ∀ y ∈ KK ( B, D ) . Suppose that for two algebras, A and B , there are elements α ∈ KK ( A ⊗ B, C ), β ∈ KK ( C , A ⊗ B ), with the property that β ⊗ A α = 1 B ∈ KK ( B, B ), β ⊗ B α = 1 A ∈ KK ( A, A ) . Then wesay that we have KK-duality isomorphisms between the K-theory (K-homology) of the algebra A and the K-homology (K-theory) of the algebra BK ∗ ( A ) ∼ = K ∗ ( B ) , K ∗ ( A ) ∼ = K ∗ ( B ) . (3.8)In fact, the algebras A and B are Poincar´e dual [37], but generally speaking these algebras arenot KK-equivalent. We now review the concept of K-amenable groups [15]. Let G be a connected Lie group and K a maximal compact subgroup. We also assume that dim ( G/K ) is even and
G/K admitsa G − invariant Spin C − structure. The G − invariant Dirac operator D := γ µ ∂ µ on G/K is afirst order self-adjoint, elliptic differential operator acting on L − sections of the Z − gradedhomogeneous bundle of spinors S . Let us consider a 0 th order pseudo-differential operator O = D (1 + D ) − acting on H = L ( G/K, S ). C ( G/K ) acts on H by multiplication of operators. G acts on C ( G/K ) and on H by left translation, and O is G − invariant. Then, the set ( O , H, X )defines a canonical Dirac element α G = KK G ( C ( G/K ) , C ). Theorem 3.3 (
G. Kasparov [38])
There is a canonical Mishchenko element α G ∈ KK G ( C ( G/K ) , C ) (3.9) such that the following intersection products occur: T . α G ⊗ C β G = 1 C ( G/K ) ∈ KK G ( C ( G/K ) , C ( G/K )) . T . β G ⊗ C ( G/K ) α G = γ G = KK G ( C , C ) where γ G is an element in KK G ( C , C ) . All solvable groups are amenable, while any non-compact semisimple Lie group is non-amenable.For a semisimple Lie group G or for G = R n , a construction of the Mishchenko element β G canbe found in [15]. We now come to the basic theorem and definition: Theorem 3.4 (
G. Kasparov [38])
If group G is amenable, then γ G = 1 . Definition 3.3
A Lie group G is said to be K-amenable if γ G =1. Proposition 3.1
The following statements hold: . Any solvable Lie group and any amenable Lie group is K-amenable. P . The non-amenable groups SO ( n, are K-amenable Lie groups [16] . P . The groups SU ( n, are K-amenable Lie groups [39] . P . The class of K-amenable groups is closed under the operations of taking subgroups, underfree and direct products [40] . C ∗ − algebras Twisted crossed products of C ∗ − algebras . Let us consider a general family of twistedactions of locally compact groups on C ∗ − algebras, and the corresponding twisted crossed prod-uct C ∗ − algebras. We start with the definition of a twisted action of a locally compact group G on C ∗ − algebra A (see for detail Ref. [12]). Let Aut A and U M ( A ) denote its automorphismgroup and the group of unitary elements in its multiplier algebra M ( A ). A twisted actionof G on A is a pair α, u of Borel maps α : G → Aut
A , u : G × G → U M ( A ) satisfying α s ◦ α t = Ad u ( s, t ) ◦ α st , α r ( u ( s, t )) u ( r, st ) = u ( r, s ) u ( rs, t ) . These twisted actions and a twistedBanach ∗− algebra L ( A, G, α, u ) have been introduced in [41]. The quadruple (
A, G, α, u ) canbe refered as a (separable) twisted dynamical system : • The covariant theory of the system (
A, G, α, u ) can be realized on Hilbert space. The corre-sponding reduced crossed product can be defined as the C ∗ − algebra generated by the regularrepresentation [12]. In [42] a duality theorem has been proved for that C ∗ − algebra. • The twisted cross product A × α,u G was defined as a C ∗ − algebra whose representation theoryis the same as the covariant representation theory of ( A, G, α, u ) on Hilbert space [12]. A crossproduct by coactions of possibly non-amenable groups also has been considered in [43]. • Suppose that ( α, u ) is a twisted action of an amenable group G on a C ∗ − algebra A which is thealgebra of sections of a C ∗ − bundle E over X ; each α leaves all ideals I x = { a ∈ A : a ( x ) = 0 } invariant. It has been shown [13] that A × α,u G is the algebra of sections of a C ∗ − bundleover X with fibers of the form ( A/I x ) × α ( x ) ,u ( x ) G . This result has been proved by using thestabilization trick of [12]. Stabilisation trick : The twisted cross product algebra A ⋊ σ Γ is stably equivalent to the crossproduct ( A ⊗ K ) ⋊ Γ, where K denotes the compact operators (we refer the reader to thearticle [12] for details). Twisted K-groups . Let Γ be a discrete cocompact subgroup of a solvable simply-connectedLie group G . It has been shown in [14] that K ∗ ( C ∗ (Γ)) ∼ = K ∗ +dim G ( G/ Γ) . (4.1)For a multiplier σ on Γ (a cocycle σ ∈ Z (Γ , U (1))) the K-theory of the twisted group algebra C ∗ (Γ , σ ) is that of a continuous-trace C ∗ − algebra B σ with spectrum G/ Γ, i.e. the twisted K-theory K ∗ ( G/ Γ , δ ( B σ )) [13]. The Disxmier-Douady class δ ( B σ ) can be identified as the image It is often useful to introduce C ∗ − algebras A as involute Banach algebras for which the following equalitieshold ( xy ) ∗ = y ∗ x ∗ , || x ∗ x || = || x || for x, y ∈ A . A unique norm is given for any x by || x || = (spectral radius of( x ∗ x ) / ) σ under a homomorphism [44, 45]: δ : H (Γ , U (1)) −→ H ( G/ Γ , Z ) , (4.2)it depends only on the homotopy class of σ in H (Γ , U (1)). This result has been extended in[13] to describe the K-theory of the twisted transformation group algebras C ( X ) × τ,ω Γ, where X is a Γ − space and ω ∈ Z (Γ , C ( X, U (1))). Thus K ∗ ( C ( X ) × τ,ω Γ) is isomorphic to a twistedK-group K ∗ (( G × X ) / Γ , δ ( ω )) of the orbit space ( G × X ) / Γ for the diagonal action, and identifythe twist δ ( ω ) ∈ H (( G × X ) / Γ , Z ). Further generalisation can be obtained for describing thetwisted transformation group algebra C ( Y ) × τ,ω G associated with a locally compact group G ,a principal G − bundle Y and cocycle ω ∈ Z ( G, C ( Y, U (1))) [13].Let Γ ⊂ G be a lattice in G and A be an algebra admitting an automorphic action of Γ.The cross product algebra [ A ⊗ C ( G/K )] ⋊ Γ is Morita equivalent to the algebra of continuoussections vanishing at infinity C (Γ \ G/K, E ). Here E → Γ \ G/K is the flat A − bundle definedas the quotient E = ( A × G/K ) / Γ −→ Γ \ G/K , (4.3)and we consider the diagonal action of Γ on A × G/K . Theorem 4.1 (
G. Kasparov [38])
Let G be a K-amenable, then ( A ⋊ Γ) ⊗ C ( G/K ) and [ A ⊗ C ( G/K )] ⋊ Γ have the same K-theory. Corollary 4.1
Let G be a K-amenable, then ( A ⋊ Γ) ⊗ C ( G/K ) and C (Γ \ G/K, E ) have thesame K-theory. It means that for ℓ = 0 , one has K ℓ ( C (Γ \ G/K, E )) ∼ = K ℓ +dim( G/K ) ( A ⋊ Γ) . (4.4)We follow the lines of the article [15] in the formulation and in the proof of the main theoremwhich generalizes theorems of [12] and [13]. Theorem 4.2 (
A. L. Carey, K. C. Hannabuss, V. Mathai and P. McCann [15])
Let Γ be a lat-tice in a K-amenable Lie group G and K be a maximal compact subgroup of G . Then K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ +dim( G/K ) (Γ \ G/K, δ ( B σ )) , (4.5) where σ ∈ H (Γ , U (1)) is any multiplier on Γ , K ∗ (Γ \ G/K, δ ( B σ )) is the twisted K-theory of acontinuous trace C ∗ − algebra B σ with spectrum Γ \ G/K , and δ ( B σ ) denotes the Dixmier-Douadyinvariant of B σ .Proof. First suppose that A = C and Γ acting trivially on C . Because of the Corollary 4.1 when γ G = 1 one gets (cid:26) ( A ⋊ Γ) ⊗ C ( G/K ) C (Γ \ G/K, E ) (cid:27)| {z } have the same K − theory = (cid:26) ( C ⋊ Γ) ⊗ C ( G/K ) C (Γ \ G/K, E ) (cid:27)| {z } have the same K − theory Theorem ⇐ == ⇒ (cid:26) [ C ⊗ C ( G/K )] ⋊ Γ C ( G/K, E ) (cid:27)| {z } have the same K − theory = ⇒ C ∗ (Γ) and C (Γ \ G/K ) have the same K − theory . Suppose σ ∈ H (Γ , U (1)). If G is K-amenable, then using Corollary 4.1 one sees (cid:26) ( A ⋊ Γ) ⊗ C ( G/K ) C (Γ \ G/K, E ) (cid:27)| {z } have the same K − theory = ⇒ (cid:26) ( A ⋊ σ Γ) ⊗ C ( G/K ) C (Γ \ G/K, E σ ) (cid:27)| {z } have the same K − theory , where E σ = ( A ⊗ K × G/K ) / Γ −→ Γ \ G/K .
Since by definition the twisted K-theory K ∗ (Γ \ G/K, δ ( B σ )) is the K-theory of the continuoustrace C ∗ − algebra B σ = C (Γ \ G/K, E σ ) with spectrum Γ \ G/K . Thus Eq. (4.5) follows. (cid:4) K ∗ ( δ ) ≡ K ∗ (Γ \ G/K, δ ( B σ ))( G is K-amenable) δ ( B σ ) H (Γ \ G/K, Z )non twisted group K ∗ (0)(Γ \ G/K is orientable) 0 ( B σ is Morita equivalent to C (Γ \ G/K ) Z non twisted group K ∗ (0)(Γ \ G/K is not orientable) 0 ( B σ is Morita equivalent to C (Γ \ G/K ) 0twisted group K ∗ ( δ ) = 0 ( K ∗ ( δ ) can be not isomorphic to K ∗ (0),but there is isomorphism for δ = 0 [12]) One of the main result of [15] says that for lattices in K-amenable Lie groups the reduced andunreduced twisted group C ∗ − algebras have canonically isomorphic K-theory. If σ ∈ H (Γ , U (1))is a multiplier on Γ and Γ is a lattice in a K-amenable Lie group, then the canonical morphism C ∗ (Γ , σ ) → C ∗ r (Γ , σ ) induces an isomorphism K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ ( C ∗ r (Γ , σ )) . (4.6) Corollary 4.2
Suppose G is a connected Lie group and K a maximal compact subgroup suchthat dim( G/K ) = 3 . Let Γ be an uniform lattice in G and σ ∈ H (Γ , U (1)) be any multiplier on Γ . Suppose also that G is K-amenable, then for ℓ = 0 , (mod 2) K ℓ ( C ∗ r (Γ , σ )) ∼ = K ℓ ( C ∗ r (Γ)) ∼ = K ℓ +1 (Γ \ G/K ) . (4.7)Indeed, by the main Theorem 4.2, K ℓ ( C ∗ r (Γ)) ∼ = K ℓ +dim( G/K ) (Γ \ G/K ) , for ℓ = 0 , C ∗ r (Γ , σ ) is Morita equivalent to C ⋊ Γ, andsince G is K-amenable, C ⋊ Γ ⊗ C ( G/K ) is Morita equivalent to B σ = C (Γ \ G/K, E σ ). Hereas before E σ is a locally trivial bundle of C ∗ − algebras over Γ \ G/K with fibre K . For theDixmier-Douady invariant one has δ ( B σ ) = δ ( σ ) ∈ H (Γ \ G/K, Z ) ∼ = H (Γ , Z ) . If Γ \ G/K isnot orientable, then H (Γ \ G/K, Z ) = { } . Therefore δ ( B σ ) = 0 (see Table 4.2) and Eq. (4.7)holds. On the contrary, when Γ \ G/K is orientable, δ ( σ ) = 0 for all σ ∈ H (Γ , U (1)) [15], and B σ is Morita equivalent to C (Γ \ G/K ) (see Table 4.2). In this case again we have Eq. (4.7).
Corollary 4.3 (
A. L. Carey, K. C. Hannabuss, V. Mathai and P. McCann [15])
Let M = K (Γ , be an Eilenberg-MacLane space which is a connected locally-symmetric, compact, three-dimensional manifold. Let σ ∈ H (Γ , U (1)) be any multiplier on Γ , then one has K ℓ ( C ∗ r (Γ , σ )) ∼ = K ℓ ( C ∗ r (Γ)) ∼ = K ℓ +1 ( M ) , ℓ = 0 , . (4.8)13ndeed, K (Γ ,
1) is locally symmetric (see the Appendix for the necessary information on Eilenberg-MacLane spaces), and therefore it is of the form Γ \ G/K , where G is a connected Lie group, K is a maximal compact subgroup such that dim( G/K ) = 3 and Γ ⊂ G is an uniform lattice in G .According to Thurston’s list of three-geometries or locally homogeneous spaces, one has • G = R ⋊ SO (3) , G/K = R (flat). • G = SO (3 , , G/K = H (hyperbolic; compact example: non-trivial Σ g ≥ − bundle over S . • G = SO (2 , ⋊ R , G/K = H × R (hyperbolic; compact example: trivial Σ g ≥ − bundleover S , i.e. Σ g × S ). • G = N il = Z ⋊ R (central, non-split extension), G/K = N il (flat; compact example: T − bundle over S via mapping torus). • G = S ol = R ⋊ R (split extension), G/K = S ol (flat; compact example: T − bundleover S via mapping torus). • G = ^ SO (2 , ⋊ R , G/K = ^ SO (2 ,
1) (hyperbolic; compact example: S − bundle over F g ≥ ).For all of these three-manifolds γ G = 1. More two locally homogeneous spaces in Thurston’s listare not locally symmetric. We can apply Corollary 4.2 in order to deduce Corollary 4.3. Using the mathematical tools exposed so far, we are now able to formulate new results concerningK-amenability in the four-dimensional case.
Corollary 4.4
Lie groups G associated with spaces G/K = R , H , C P , CH , H × R , H × H , H × R , N il × R , S ol , N il , S ol m,n (including S ol × R ) , S ol , ^ PSL (2 , R ) × R , F enu-merated in Table 2.3 are K-amenable.Proof . We need to proof that γ G = 1. According to list of four-geometries, Table 2.3, one hasthe following result: C . G = R ⋊ SO (4) , G/K = R , γ G = 1 since R and SO (4) are amenable, and so is theirsemidirect product (Proposision 3.1). C . G = SO (4 , , G/K = H , γ G = 1 by Kasparov’s theorem (Proposition 3.1). C . G = SU (3) , G/K = C P ≃ U (3) / ( U (1) × U (2)) ≃ SU (3) / S ( U (1) × U (2)) , γ G = 1(Proposition 3.1). C . G = G/K = CH , the geometry CH is a K¨ahlerian symmetric space and certainly carriesa complex structure, γ G = 1. C . G/K = H × R , H × H , H × R , γ G = 1 since these groups are free products of K-amenable groups (Proposition 3.1). 14 . G/K = N il × R , S ol , N il , S ol m,n (including S ol × R ) , S ol γ G = 1 since nilpotentand solvable groups are K-amenable groups and so is their semidirect product (Proposition3.1). C . G/K = ^ PSL (2 , R ) × R , group ^ PSL (2 , R ) is diffeomorphic to R , γ G = 1 since X is freeproducts of K-amenable groups (Proposition 3.1). C . G/K = F . One can choose a subgroup G X of X which admits X as a principal homoge-neous space [29]. Take G X = R ⋉ SL (2 , R ) (with natural action of SL (2 , R ) on R ) consistof the upper triangular matricies with positive diagonal entries. γ G = 1 because of thesemidirect product of K-amenable groups (Proposision 3.1). (cid:4) The other five locally homogeneous spaces S , S × R , S × S , S × H , S × R in Thurston’slist are not locally symmetric spaces. Remark 4.1
Are Corollaries 4.3 and 4.4 still valid without the locally symmetric assumptionon K (Γ , ? The answer on this interesting question was found in [15] where the interestingconjecture has been formulated without the locally symmetric assumption on three-manifolds:Suppose K (Γ , is a connected, compact, three-manifold which is an Eilenberg-MacLane spacewith fundamental group Γ . Then for any multiplier σ ∈ H (Γ , U (1)) on Γ , K j ( C ∗ r (Γ , σ )) ∼ = K j ( C ∗ r (Γ)) ∼ = K j +1 ( K (Γ , , j = 0 , . We can generalize this statement for the case of four-manifolds in terms of conjecture:
Conjecture 4.1
Let K (Γ , be a connected, compact, four-dimensional manifold which is anEilenberg-MacLane space with fundamental group Γ . For any multiplier σ ∈ H (Γ , U (1)) on Γ one has K j ( C ∗ r (Γ , σ )) ∼ = K j ( C ∗ r (Γ)) ∼ = K j +1 ( K (Γ , , j = 0 , . (4.9) First we discuss T-duality in one direction only; then T-dualizing on the torus will be consideredin the next sections. A more general case with T-dualizing on T n , n > M × S , but can also be applied locally in the case of S − fibrations over M [46],and can be generalized to situations with nontrivial NS three-form flux H . A more general casewhere X is an oriented S − bundle over the manifold M characterizes by its first Chern class c ( X ) ∈ H ( M, Z ) in the presence of (possibly nontrivial) H − flux δ ( B ) ≡ [ H ] ∈ H ( X, Z ). Remark 5.1
To simplify notations we will use the same notation for a cohomology class [ H ] ,or for a representative H , throughout the last part of this paper. For the reader it should be clearwhich is meant from the context. Probably the Dixmier-Douady invariant δ ( σ ) = 0 for all σ ∈ H (Γ , U (1)) for Γ as in the conjecture. Let us consider a bundle whose fiber F is ( p − − connected; this means that for k < p the k − th homotopygroup π k
15t has been argued [22] that the T-dual of X , b X , is again an oriented b S − bundle over c MS −−−−→ X π y M T − Duality ←−−→ b S −−−−→ b X b π yc M (5.1)supporting H − flux b H ∈ H ( b X, Z ), and c ( X ) = π ∗ H , c ( b X ) = b π ∗ b H . Here π ∗ : H k ( X, Z ) → H k − ( M, Z ), and similarly b π ∗ , denotes the pushforward maps. (As an example, at the level ofthe de Rham cohomology the pushforward maps π ∗ and b π ∗ are simply the integrations along the S − fibers ( b S − fibers ) of X ( b X ).) Proposition 5.1 (Gysin sequence) Let π : X → M be an oriented sphere bundle with fiber S k .Then there is a long exact sequence (see for example [47] ) . . . −→ H n ( X ) π ∗ −→ H n − k ( M ) ∧ e −→ H n +1 ( M ) π ∗ −→ H n +1 ( M ) −→ . . . (5.2) where the maps π ∗ , ∧ e , and π ∗ are an integration along the fiber, a multiplication by the Eulerclass, and the natural pullback, respectively. In the case of an oriented S bundle with first Chern class c ( X ) = F ∈ H ( M, Z ) one gets . . . −→ H ℓ ( M, Z ) π ∗ −→ H ℓ ( X, Z ) π ∗ −→ H ℓ − ( M, Z ) F ∪ −→ H ℓ +1 ( M, Z ) −→ . . . (5.3)In particular, for a two-dimensional base manifold M , the Gysin sequence gives an isomorphismbetween H ( X, Z ) and H ( M, Z ), i.e. between Dixmier-Douady classes on X and line bundleson M . For example S manifold can be considered as an S − bundle over S by means of theHopf fibration. T-duality in the absence of H − flux leads to S × S manifold supported by oneunit of H − flux [22]. Circle bundles on Riemannian surfaces . Let us consider twisted K-groups of circlebundles over two-manifolds and their T-duals. It can be shown that in this case K of eachspace is related to K of its dual. This class of manifolds includes: • The familiar examples of NS5-branes . • Three-dimensional Lens spaces . • N il − manifolds .The K-groups determined by the Atiyah-Hirzebruch spectral sequence. It is convenient to con-sider the first differential d = Sq + H of the sequence only. If H even ( X, Z ) and H odd ( X, Z ) arethe even and odd cohomology classes of the manifold X , then the twisted K-groups are K ( X, H ) = Ker( H ∪ : H even −→ H odd ) H ∪ H odd ( X, Z ) , K ( X, H ) = Ker( H ∪ : H odd −→ H even ) H ∪ H even ( X, Z ) . (5.4)The case of noncommutative D2-branes has been consider in [15]. These branes can wraptwo-dimensional manifolds, which in the presence of a constant B − field are described by non-commutative Riemann surface. Let M = Σ g ≡ H / Γ g be a Riemann surface of genus g . We canspecialize to the case when G = R , K = { e } and g = 1, with Γ being Z . Let σ ∈ H (Γ σ , U (1))be any multiplier on Γ σ . The graded groups are given by Gr ( K (Σ g )) = ⊕ j E j ∞ (Σ g ) , Gr ( K (Σ g )) = ⊕ j E j +1 ∞ (Σ g ) . (5.5)16n two dimensions the Chern character is an isomorphism over the integers and therefore we get K (Σ g ) ∼ = H (Σ g , Z ) ⊕ H (Σ g , Z ) ∼ = Z , K (Σ g ) ∼ = H (Σ g , Z ) ∼ = Z g . (5.6)Using Theorem 4.2 we have K ℓ ( C ∗ r (Γ g )) ∼ = K ℓ (Σ g ) , K ℓ ( C ∗ r (Γ g , σ )) ∼ = K ℓ (Σ g , δ ( B σ )) , ℓ = 0 , . (5.7)Here B σ = C (Σ g , E σ ). Note that E σ is a locally trivial flat bundle of C ∗ − algebras over Σ g ,with fibre K ( K are compact operators), it has a Dixmier-Douady invariant δ ( B σ ) which canbe viewed as the obstruction to B σ being Morita equivalent to C (Σ g ) [15]. It is evident that δ ( B σ ) = δ ( σ ) ∈ H (Σ g , Z ) = 0 . Thus B σ is Morita equivalent to C (Σ g ) and finally K j ( C ∗ r (Γ g , σ )) ∼ = K j ( C ∗ r (Γ g )) ∼ = K j (Σ g ) ,K (Σ g ) ∼ = H (Σ g , Z ) ⊕ H (Σ g , Z ) ∼ = Z , K (Σ g ) ∼ = H (Σ g , Z ) ∼ = Z g . (5.8) Three-cycles . For a Riemann surface of genus g , H (Σ g , Z ) = Z and topologically circlebundles are classified by an integer j . The cohomology of the total space X are • j = 0 (trivial line bundle): H ( X , Z ) = Z , H ( X , Z ) = Z g +1 , H ( X , Z ) = Z g +1 , H ( X , Z ) = Z ; • j = 0 (the Chern class equal to j ): H ( X , Z ) = Z , H ( X , Z ) = Z g , H ( X , Z ) = Z g ⊕ Z j , H ( X , Z ) = Z .Then untwisted ( H = 0) and twisted ( H = k ) K-groups are given by [22]: K ( X , H = 0) = H ( X , Z ) ⊕ H ( X , Z ) = (cid:26) Z g +2 if j = 0 , Z g +1 ⊕ Z j if j = 0 ,K ( X , H = 0) = H ( X , Z ) ⊕ H ( X , Z ) = (cid:26) Z g +2 if j = 0 , Z g +1 if j = 0 .K ( X , H = k ) = H ( X , Z ) = (cid:26) Z g +1 if j = 0 , Z g ⊕ Z j if j = 0 ,K ( X , H = k ) = H ( X , Z ) ⊕ H ( X , Z ) /kH ( X , Z ) = (cid:26) Z g +1 ⊕ Z k if j = 0 , Z g ⊕ Z k if j = 0 . (5.9)In fact T-duality is the interchange of j and k . Results in the twisted K-groups K ( X , H )and K ( X , H ) being interchanged, which corresponds to the fact that RR fieldstrengths areclassified by K ( X , H ) in type IIA string theory and by K ( X , H ) in IIB. This means thatapplying the isomorphism between the two K-groups one can find the new RR fieldstrengthsfrom the old ones. Indeed one simply interchanges the Z g between H and H and the rest ofthe cohomology groups are swapped H ↔ H , H ↔ H . Lens spaces . Let us consider the case of a linking two-sphere. It gives an isomorphism ofthe twisted K-theories of Lens spaces L (1 , p ) = S / Z p (which is the Eilenberg-MacLane space,see Remark 6.1) [22]: K ℓ ( L (1 , j ) , H = k ) ∼ = K ℓ +1 ( L (1 , k ) , H = j ) . (5.10) Example [15]: In the flat case and for the Euclidean group G = R n ⋊ SO (2 n ), K = SO (2 n ), and Γ ⊂ G isa Bieberbach group, i.e. Γ is a uniform lattice in G . Also a generalization of noncommutative flat manifolds canbe defined by regarding C ∗ (Γ , σ ) as such an object. In addition σ is any group two-cocycle on Γ, due to the factthat K ∗ ( C ∗ (Γ , σ )) ∼ = K ∗ (Γ \ G/K ) . L (1 , p ) = S / Z p is the total space of the circle bundle over the two-sphere withChern class equal to p times the generator of H ( S , Z ) ∼ = Z . In particular, L (1 ,
1) = S and L (1 ,
0) = S × S . The Lens space L (2 , j ) can be considered as the nonsingular quotient X = S / Z j when j = 0 and X = C P × S when j = 0. Integral cohomology groups of theoriented Lens space L ( n, q ) are [47]: H ∗ ( L ( n, q ) , Z ) = Z in dimension 0 , Z q in dimensions 2 n , , H ≤ p ≤ ( C P × S ) = Z , (5.11) H ( L (2 , j ) , Z ) = H ( L (2 , j ) , Z ) = Z , H ( L (2 , j ) , Z ) = H ( L (2 , j ) , Z ) = Z j . A nontrivial quotient of string solution
AdS × S can be associated with circle bundles on C P (see for detail [22]). It is easy to see that H − flux is only possible for the trivial bundle j = 0, asthe nontrivial bundles have trivial third cohomology. T-duality relates the trivial bundle with H = j to the bundle with first Chern class j and no flux [22]. Remark 5.2
The case j = 1 of the T-duality mentioned above has been studied in [48] , andit was observed that the spacetime on the IIA side is not Spin − manifold, making the dualityquite nontrivial. T-dualities considered in this section are interesting because IIB string theoryon AdS × S is well understood. Note that the resulting RR fluxes are easily computed. Indeed,following the lines of [22] we can start with N units of G − flux supported on L (2 , j ) in IIB.Then in IIA theory there will be N units of G − flux supported on C P and j units of H − fluxsupported on H ( C P , Z ) ⊗ H ( S , Z ) . R P × T Remark 5.3
Note that factors Z g do not play important role in what follows. In fact, we canignore them and consider the two-sphere Σ g =0 = S or two-torus Σ g =1 = T cases. Proposition 5.2
Let M be a contractible. For freely acting of a group G on M , H n ( G, A ) = H n ( X, A ) , and H n ( G, A ) = H n ( X, A ) , where A is a trivial G − bundle and X ≡ G \ M is theorbit space G in M , providing factor topology and canonic mapping π from M to X . Let as before G be a connected Lie group, K be a maximal compact subgroup and Γ be a closeddiscrete subgroup without torsion. (The Γ − acting on G/K is defined by γ ( gK ) = ( γg ) K .)In accordance with Proposition 5.2 H n (Γ , A ) = H n (Γ \ G/K, A ), H n (Γ , A ) = H n (Γ \ G/K, A ),where A is a trivial Γ − module. As an example, let G = R n , Γ = Z n , and K = { } . Then X = T n , and H p ( T n , Z ) = H p ( T n , Z ) = Z np . (5.12)The nontrivial classes for T are H ( T , Z ) = H ( T , Z ) = Z H ( T , Z ) = H ( T , Z ) = Z H ( T , Z ) = H ( T , Z ) = Z H ( T , Z ) = H ( T , Z ) = Z K ( T ) = H ( T , Z ) ⊕ H ( T , Z ) = Z ⊕ Z K ( T ) = H ( T , Z ) ⊕ H ( T , Z ) = Z ⊕ Z K ( T ) = H ( T , Z ) ⊕ H ( T , Z ) = Z ⊕ Z K ( T ) = H ( T , Z ) ⊕ H ( T , Z ) = Z ⊕ Z (5.13)18orsion and (co)homology of R P n are H n ( R P n , Z ) = (cid:26) Z , if n odd Z , if n even H n ( R P n , Z ) = (cid:26) Z , if n odd0 , if n even H k ( R P n , Z ) = Z , , Z , , Z , . . . H k ( R P n , Z ) = Z , Z , , Z , , . . . , k = 0 , . . . , n. (5.14)Therefore torsion, (co)homology and lower K-groups of R P become H ( R P , Z ) = Z H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = 0 H ( R P , Z ) = Z H ( R P , Z ) = Z K ( R P ) = Z K ( R P ) = Z ⊕ Z K ( R P ) = Z ⊕ Z K ( R P ) = Z (5.15)The K-theories of R P × T can be combined in order to find the K-theory of the product usingthe K¨unneth formula H n ( M × M ) = M j ( H j ( M ) ⊗ H n − j ( M )) M M j Tor( H j ( M ) , H n − j − ( M )) . (5.16) Remark 5.4
Let M be a topological space (or finite CW-complex). Because of the K¨unnethformula one has K ℓ ( T n ) ∼ = K ℓ ( S ) ⊕ n − ∼ = Z ⊕ n − for ℓ = 0 , . Thus the following isomorphismsare valid: K ( M × T n ) ∼ = (cid:16) e K ( M ) ⊕ K ( M ) ⊕ Z (cid:17) ⊕ n − , K ( M × T n ) ∼ = (cid:16) K ( M ) ⊕ e K ( M ) ⊕ Z (cid:17) ⊕ n − = ⇒ K ( M × T n ) ∼ = K ( M × T n ) . (5.17)The isomorphism (5.17) is in fact the T-duality and describes a relationship between Type IIBand Type IIA D-branes on the spacetime M × T n . This isomorphism exchanges wrapped D-branes with unwrapped D-branes. In addition the powers of 2 n − give the expected multiplicityof D p − brane charges arising from wrapping all higher stable D-branes on various cycles of thetorus T n . Remark 5.5
For the compactification of type II (or type I) on an n − torus T n one can get thefollowing isomorphisms [49] : [ K ( KO )]( M × T n , T n ) = n M k =0 [ e K ( g KO )] − k ( M ) ⊕ ( nk )= [ e K ( g KO )]( M ) ⊕ n − ⊕ [ K ( KO )] − ( M ) ⊕ n − ∼ = [ K ( KO )] − ( M × T n , T n ) . (5.18) If n = 1 then under the isomorphism (5.18) of K-groups, e K ( M ) ⊗ Z K ( S ) maps to e K ( M ) ⊗ Z K − ( S ) with the summands K ( S ) and K − ( S ) interchanged. Thus, T-duality exchangeswrapped and unwrapped D-brane configurations. If n > , then (5.18) gives the anticipateddegeneracies n − of brane charges arising from the higher supersymmetric branes wrapped onvarious cycles of the torus T n . We use notation KO for group K R . R P × T is just the tensor product of the original K-theories because the K-theoryof the torus has no torsion and so the Tor term in the K¨unneth formula is trivial K ( R P × T ) = (( K ( R P ) ⊗ K ( T )) ⊕ ( K ( R P ) ⊗ K ( T )) = Z ⊕ Z ⊕ Z ,K ( R P × T ) = (( K ( R P ) ⊗ K ( T )) ⊕ ( K ( R P ) ⊗ K ( T )) = Z ⊕ Z ⊕ Z ,K ( R P × T ) = (( K ( R P ) ⊗ K ( T )) ⊕ ( K ( R P ) ⊗ K ( T )) = Z ⊕ Z ⊕ Z ,K ( R P × T ) = (( K ( R P ) ⊗ K ( T )) ⊕ ( K ( R P ) ⊗ K ( T )) = Z ⊕ Z ⊕ Z . (5.19)The universal coefficient theorem yields to the K-cohomology groups K ( R P × T ) = K ( R P × T ) = Z ⊕ Z ⊕ Z . (5.20)The untwisted K-theories associated with real projective spaces R P k +1 can be obtained identi-cally. It is known that in type IIA string theory D-branes are classified by a group K while RRfield strengths are classified by a group K . In case of R P × T both groups are Z ⊕ Z ⊕ Z .Therefore D-branes are classified by K ( R P × T ) and the physical interpretation of the gen-erators and branes wraps the T and the R P are similar to the description given in [24]. C P × Σ g × T Untwisted K-theory for the case M = C P × Σ g × T . Here we dualize the R P × X casewith no H − flux. Let the manifold X admit free cycle actions. In particular, both manifolds X and R P are circle bundles over space the Riemannian two-space Σ g and the complex projectivespaces C P respectively S −−−−→ X π y Σ g b S −−−−→ R P b π y C P (5.21)The nontrivial cohomology classes of the base spaces are H ( C P , Z ) = H ( C P , Z ) = H ( C P , Z ) = H ( C P , Z ) = Z . (5.22)We can T-dualize both circle fibers following the lines of [22, 24]. Note that T-duality exchangesthe integrals of the H − fluxes over the circle fibers with the Chern classes. Since the original H − flux vanishes the dual Chern classes also vanish. Therefore the dual spacetime c M consistsof the product of two trivial circle bundles over the original base c M = C P × Σ g × S α × S β , (5.23)where α and β associated with the Chern classes c ∈ H (Σ g , Z ) and c ∈ H ( C P , Z ) of thebundles over Σ g and C P respectively. The dual H − flux, b H , is the sum of two H − fluxes thatintegrate to the Chern classes. Twisted K-theory for the case c M = C P × Σ g × S α × S β . The twisted K-theory of c M ,as the theory which has been performed due to an even number of T-dualities, must agree withthe untwisted K-theory of the space R P × X . The K¨unneth formula for K-homology is0 −→ K ∗ ( M ) ⊗ K ∗ ( M ) −→ K ∗ ( M × M ) −→ Tor( K ∗ ( M ) , K ∗ ( M )) −→ X with coefficients in Z can be computed using the result Eq. (5.9) and theuniversal coefficient theorem. 20 heorem 5.1 (Universal Coefficient Theorem) For any space X and associated abelian group G the following result holds: T . The homology group of X with coefficients in G has a splitting: H p ( X ; G ) ≃ H p ( X ) ⊗ G ⊕ Tor ( H p − ( X, G )) . T . The cohomology group of X with coefficients in G has a splitting: H p ( X ; G ) ≃ Hom ( H p ( X ) , G ) ⊕ Ext ( H p − ( X ) , G ) . Remark 5.6
The (splittings) isomorphisms given by the universal coefficient theorem are saidto be unnatural isomorphisms. The following maps of exact sequences are natural: −→ H p ( X ) ⊗ G −→ H p ( X ; G ) −→ Tor( H p − ( X ) , G ) −→ −→ H p ( X, Z ) ⊗ G −→ H p ( X ; G ) −→ Tor( H p +1 ( X ; Z ) , G ) −→ ←− Hom( H p ( X ) , G ) ←− H p ( X ; G ) ←− Ext( H p +1 ( X ) , G ) ←− G = Z ) we have the following result: Corollary 5.1
For any space X for which H ℓ ( X ) and H ℓ − ( X ) are finite generated Z − modulesit follows H ℓ ( X ) ≃ F ℓ ( X ) ⊕ Tor( H ℓ − ( X )) . (5.25) Here F ℓ ( X ) and Tor( H ℓ − ( X )) are the free and torsion parts of H ℓ ( X ) and H ℓ − ( X ) , respec-tively. We can evaluate the homology of the product using the K¨unneth formula. It gives K ( R P × X ) = (cid:0) K ( R P ) ⊗ K ( X ) (cid:1) ⊕ (cid:0) K ( R P ) ⊗ K ( X ) (cid:1) ⊕ Tor (cid:0) K ( R P ) , K ( X ) (cid:1) ⊕ Tor (cid:0) K ( R P ) , K ( X ) (cid:1) ,K ( R P × X ) = (cid:0) K ( R P ) ⊗ K ( X ) (cid:1) ⊕ (cid:0) K ( R P ) ⊗ K ( X ) (cid:1) ⊕ Tor (cid:0) K ( R P ) , K ( X ) (cid:1) ⊕ Tor (cid:0) K ( R P ) , K ( X ) (cid:1) . (5.26) Remark 5.7
Note that
Tor(
A, B ) vanishes unless both A and B contain torsion components.If m and n are positive integers, then Tor( Z m , Z n ) = Z ( m,n ) , where ( m, n ) denotes the greatestcommon divisor of m and n . K ( R P × X ; H = 0 , j = 0) = (cid:0) Z g +2 ⊗ Z (cid:1) ⊕ (cid:0) Z g +2 ⊗ ( Z ⊕ Z ) (cid:1) ⊕ ⊕ Z g +3 ⊕ Z ,K ( R P × X ; H = 0 , j = 0) = (cid:0) ( Z g +1 ⊕ Z j ) ⊗ Z (cid:1) ⊕ (cid:0) Z g +1 ⊗ ( Z ⊕ Z ) (cid:1) ⊕ Tor( Z j , Z ) ⊕ Z g +2 ⊕ Z j ⊕ Z ⊕ Z ( j, ,K ( R P × X ; H = k, j = 0) = (cid:0) Z g +1 ⊗ Z (cid:1) ⊕ (cid:0) ( Z g +1 ⊕ Z k ) ⊗ ( Z ⊕ Z ) (cid:1) ⊕ ⊕ Z g +2 ⊕ Z k ⊕ Z ⊕ Z ( k, ,K ( R P × X ; H = k, j = 0) = (cid:0) ( Z g ⊕ Z j ) ⊗ Z (cid:1) ⊕ (cid:0) ( Z g ⊕ Z k ) ⊗ ( Z ⊕ Z ) (cid:1) ⊕ Tor( Z j , Z ) ⊕ Z g +1 ⊕ Z j ⊕ Z k ⊕ Z ⊕ Z ( j, ⊕ Z ( k, ,K ( R P × X ; H = 0 , j = 0) = (cid:0) Z g +2 ⊗ ( Z ⊕ Z ) (cid:1) ⊕ (cid:0) Z g +2 ⊗ Z (cid:1) ⊕ ⊕ Z g +3 ⊕ Z ,K ( R P × X ; H = 0 , j = 0) = (cid:0) ( Z g +1 ⊕ Z j ) ⊗ ( Z ⊕ Z ) (cid:1) ⊕ (cid:0) Z g +1 ⊗ Z (cid:1) ⊕ ⊕ Z g +2 ⊕ Z j ⊕ Z ⊕ Z ( j, ,K ( R P × X ; H = k, j = 0) = (cid:0) Z g +1 ⊗ ( Z ⊕ Z ) (cid:1) ⊕ (cid:0) ( Z g +1 ⊕ Z k ) ⊗ Z (cid:1) ⊕ ⊕ Tor( Z k , Z )= Z g +2 ⊕ Z k ⊕ Z ⊕ Z ( k, ,K ( R P × X ; H = k, j = 0) = (cid:0) ( Z g ⊕ Z j ) ⊗ ( Z ⊕ Z ) (cid:1) ⊕ (cid:0) ( Z g ⊕ Z k ) ⊗ Z (cid:1) ⊕ ⊕ Tor( Z k , Z )= Z g +1 ⊕ Z j ⊕ Z k ⊕ Z ⊕ Z ( j, ⊕ Z ( k, . (5.27)In the absence of Neveu-Schwarz (NS) flux, K group describes RR field strengths in IIB stringtheory, which we write locally as G p = dG p − . It has been suggested that RR field strengthsare classified by twisted K-theory. Field strengths satisfy d G p = ( Sq + H ∪ ) G p = 0 . Here Sq is the Steenrod squares that takes torsion p − classes to torsion ( p + 3) − classes. In the case ofS-duality, a configuration in which the G − flux valued in Z ⊂ H ( R P × X ) is nonzero, we canfind another configuration which corresponds to a class in the K-theory twisted by the original G . Let us consider dualizing of R P × T with no H − flux, example from Section 5.1. In thiscase we are looking for a trivial G − bundle (see Proposition 5.2), Σ g =1 = T and X = T .Then, T-dualizing the circles becomes b S −→ R P b π y S × T π y C P × T × T − Duality ←−−→ C P × Σ g × S α × S β | {z } H =0 2T − Dualities ←−−→ R P × T | {z } H =0 (5.28)The twisted K-theory associated with space c M = C P × Σ g × S α × S β , as we have performedan even number of T-dualities, must agree with the untwisted K-theory of the original space In the classical limit, type II supergravity, we can forget the flux quantization condition and look at realcohomology. In fact, we can no longer see the Sq term. This theory contains RR potentials C p − , a NS three-form H and a gauge-invariance C p − → C p − + d Λ p − + H ∧ Λ p − , for any set of forms Λ k . It follows that thereare two natural field strengths G p = dC p − and F p = dC p − + H ∧ C p − . In addition, G p is closed and F p isgauge-invariant. P × X : b S −→ R P b π y S → X π y C P × Σ g × T − Duality ←−−→ C P × Σ g × S α × S β | {z } H =0 2T − Dualities ←−−→ R P × X | {z } H =0 (5.29) Definition 6.1
Let Γ be a discrete group. A based topological space (or CW complex) X iscalled an Eilenberg-MacLane space of type K (Γ , n ) , where n ≥ , if Γ k ( X ) = Γ for k = n and Γ k ( X ) = 0 for k = n . It means that all the homotopy groups Γ k ( X ) are trivial except for Γ n ( X ) ,which is isomorphic to Γ . Note that K (Γ ,
0) is a CW complex with Γ = Γ having contractible components. For n = 1 andfinitely generated group Γ the spaces K (Γ ,
1) are well known: K ( Z ,
1) = S , K ( Z ,
1) = R P ∞ ,and K ( Z m ,
1) = L ∞ m for m >
2. For n ≥ n ≥ K (Γ , n ) which can be constructedas a CW complex. In general they are more complicated objects which play a fundamentalrole in the connection between homotopy and (co)homology.
Remark 6.1
Recall that the simplest examples of Eilenberg-MacLane spaces are: R . K ( Z , ∼ C P ∞ = S ∞ /S ; H ∗ ( C P ∞ ; Z ) ≃ Z p [ t ] , t ∈ H ( C P ∞ ; Z ) for all primes p ≥ . R . K ( Z p n , ∼ S ∞ / Z p n = lim N →∞ L N +1 p n (1 , , ..., . The general lens space L n − m ( q , ..., q n − ) of dimension n − , is defined as the orbit space of the sphere S n − ⊂ C n under the actionof the group Z m given by ( z , ..., z n ) ( e π √− /m z , e π √− q /m z , ..., e π √− q n − /m z n ) , where each q ℓ is relatively prime to m . With this action of Z m we set L n − m ( q , ..., q n − ) = S n − / Z m . In particular, K ( Z ,
1) = R P = lim N →∞ R P N . R . Γ = F (a free group), K ( F, ∼ S ∨ . . . ∨ S (a bouquet of circles). Any two Eilenberg-MacLane spaces of K (Γ , n ) are weakly homotopy equivalent. Also K (Γ , n )is a homotopy commutative H − space. The following result holds: K (Γ × Γ , n ) = K (Γ , n ) × K (Γ , n ) , Ω( K (Γ , n )) = K (Γ , n − , (6.1)where Ω( K (Γ , n )) is the loop space relative to some base point. The second equation is essentiallya consequence of the homotopy groups isomorphism π ℓ (Ω( X )) ≃ π ℓ +1 ( X ). Theorem 6.1 (Hurewicz Isomorphism Theorem) For n − connected cell complex C the groups π n +1 ( C ) and H n +1 ( C ; Z ) ( n > ) are isomorphic. The Whitehead theorem implies that there is a unique K (Γ , n ) space up to homotopy equivalence in thecategory of topological spaces of the homotopy type of a CW complex.
23y Hurwitz’s Theorem 6.1, H n ( K (Γ , n ); Z ) ≃ π n ( K (Γ , n )) ≃ Γ. It follows that there is a canon-ical isomorphism H n ( K (Γ , n ); G ) ≃ Hom (Γ , G ) , where G is any abelian group and Hom (Γ , G )denotes the additive group of homomorphisms from the abelian group Γ to G . Partial computing of H ∗ ( K (Γ , n ) , Q ) ring . Let Γ be a finitely generated abelian groupand let Γ = Z ⊕ . . . ⊕ Z ⊕ G , where G is a finite group. Then, respectively, K (Γ , n ) = K ( Z , n ) ×· · · × K (Γ , n ) × K ( G, n ) . Because of the K¨unneth formulae we get H ∗ ( K (Γ , n ); Q ) = H ∗ ( K ( Z , n ); Q ) ⊗ . . . ⊗ H ∗ ( K ( Z , n ); Q ) ⊗ H ∗ ( K ( G, n ); Q ) . (6.2)Note that if integer cohomologies of a topological space are finite then its rational cohomolo-gies are trivial; H ∗ ( K ( G, n ); Q ) = H ∗ ( pt ; Q ). Therefore we have to compute H ∗ ( K ( Z , n ); Q ),neglecting the last term in Eq. (6.2).Let k be a field, Λ k ( x , ..., x n ) (dim x = n ) external algebra generated by x , ..., x n , i.e. k − algebra with generators x ℓ and relations x k x ℓ = − x ℓ x k , x ℓ = 0. The dimension of thisalgebra is 2 m and its basis form monoms x ℓ . . . x ℓ s , 1 ≤ ℓ < ... < ℓ s ≤ m . Finally, k [ x , ..., x m ]denotes an algebra of polynomials with coefficients in k . Theorem 6.2 H ∗ ( K ( Z , n ); Q ) = (cid:26) Λ Q ( x ) for odd n , Q [ x ] for even n . (6.3) The formula H ∗ ( K ( Z , n ); Q ) = Λ Q ( x ) (dim x = n ) has a simple meaning: H ∗ ( K ( Z , n ); Q ) = H ∗ ( S n ; Q ) . The formula H ∗ ( K ( Z , n ); Q ) = Q [ x ] (dim x = n ) means that H ℓ ( K ( Z , n ); Q ) = Q for ℓ = 0 , n, n, ..., H ℓ ( K ( Z , n ); Q ) = 0 for other value of ℓ and the element x q , = x ∈ H n ( K ( Z , n ); Q ) , generates H qn ( K ( Z , n ); Q ) over k . Corollary 6.1
Let rank Γ = r , then H ∗ ( K (Γ , n ); Q ) = (cid:26) Λ Q ( x , ..., x r ) for odd n , Q [ x , ..., x r ] for even n . (6.4)Let us consider the Eilenberg-MacLane complexes K (Γ , n ) for Γ = Z p n and p prime. We arelooking for all cohomological operations modulo an arbitrary prime p . Remind that cohomologiesof the space K ( Z p ,
1) = L ∞ p are known: H q ( K ( Z p , Z ) = Z p for even q , and H q ( K ( Z p , Z ) =0 for odd q . For positive q there is a multiplicative isomorphism: H ∗ ( K ( Z p , Z ) ∼ = Z p [ x ] , x ∈ H ( K ( Z p , Z ) . (6.5) Theorem 6.3 (A. T. Fomenko and D. B. Fuks [50] ) For < q ≤ n + 4 p − the ring H q ( K ( Z , n ); Z p ) isomorphic to H q ( K ( Z p , n − Z ) . For further interesting examples of computing of the groups H ∗ ( K (Γ , n ) , Q ) we refer the readerto the book [50]. Acknowledgements
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