T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group

We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a “classical” T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be “non-classical,” that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part of the paper contains a detailed analysis of the classifying space for topological T-duality, as well as the T-duality group and its action. The issue of possible non-uniqueness of T-duals can be studied via the action of the T-duality group. T-duality is a duality of type II string theories that involves exchanging a theory compactified on a torus with a theory compactified on the dual torus. The T-dual of a type II string theory compactified on a circle, in the presence of a topologically nontrivial NS 3-form H-flux, was analyzed in special cases in [1, 2, 19, 28, 30]. There it was observed that T-duality changes not only the H-flux, but also the spacetime topology. A general formalism for dealing with T-duality for compactifications arising from a free circle action was developed in [4]. This formalism was shown to be compatible with two physical constraints: (1) it respects the local Buscher rules [10, 11], and (2) it yields an isomorphism on twistedK-theory, in which the Ramond-Ramond charges and fields take their values [32, 45, 46, 37, 7, 39]. It was shown in [4] that T-duality exchanges the first Chern class with the fiberwise integral of the H-flux, thus giving a formula for the T-dual spacetime topology. The purpose of this paper is to extend these results to the case of torus bundles of higher rank. It is common knowledge that noncommutative tori occur naturally in string theory and in M-theory compactifications [44, 13]. This paper derives another instance where they make a natural appearance, in the sense that if we start with a classical spacetime that is a principal torus bundle with H-flux, then the T-dual is sometimes a continuous field of noncommutative tori, and we characterize exactly when this happens. The first part of the paper forms a sequel to our earlier paper [33] on T-duality for principal torus bundles with an H-flux. In that paper, we dealt primarily with the case of fibers which are tori of dimension 2, in VARGHESE MATHAI AND JONATHAN ROSENBERG 3 which case every principal bundle is T-dualizable, though not necessarily in the “classical” sense; in fact, if the integral of the H-flux over the torus fibers is non-trivial in cohomology, then the T-dual of such a principal torus bundle with H-flux is a continuous field of noncommutative tori. A similar phenomenon was also noticed in [31], in fact for one of the same examples studied in [33] (a trivial T 2-bundle over S1, but with non-trivial flux). In this paper, we consider principal torus bundles of arbitrary dimension. Still another phenomenon appears: there are some principal bundles with H-flux which are not dualizable even in our more general sense (where the T-dual is allowed to be a C∗-algebra). The simplest case of this phenomenon is the 3-dimensional torus, considered as a 3-torus bundle over a point, with H-flux chosen to be a non-zero integer multiple of the volume 3-form on the torus. More precisely, a given principal torus bundle with H-flux, over a base Z, is T-dualizable in our generalized sense if and only if the restriction of the H-flux to a torus fiber T is trivial in cohomology. Moreover, the T-dual of such a principal torus bundle with H-flux is “non-classical” if and only if the push-forward of the flux in H1(Z,H2(T )) is non-trivial. In [6], the results of [33] and of this paper were applied and extended, and it was shown that every principal torus bundle is T-dualizable in an even more general sense, where the T-dual is a field of non-associative tori, i.e., taking us out of the category of C∗-algebras! The other main part of this paper contains the analysis of the classifying space for topological T-duality and of the T-duality group, as well as its action. The theory divides naturally into two cases, when there is a classical T-dual, and when there is only a non-classical T-dual, and the T-duality group respects these two cases. One somewhat unexpected result is that, when n = 2, the classifying space for torus bundles with H-flux splits as a product, one factor corresponding to the classical case and one factor corresponding to the non-classical case. For a rank n torus bundle with H-flux, the T-duality group is GO(n, n;Z), and acts by homotopy automorphisms of the classifying space for classically dualizable bundles. Study of this group enables us to understand puzzling instances of non-uniqueness of T-duals. Some of the results of this paper were announced in [34]. 1 Notation and review of results from [33] We begin by reviewing the precise mathematical framework from [33]. We assume X (which will be the spacetime of a string theory) is a (secondcountable) locally compact Hausdorff space. In practice it will usually be a compact manifold, though we do not need to assume this. But we assume 4 T-DUALITY VIA NONCOMMUTATIVE TOPOLOGY, II that X is finite-dimensional and has the homotopy type of a finite CWcomplex, in order to avoid some pathologies. We assume X comes with a free action of a torus T ; thus (by the Gleason slice theorem [24]) the quotient map p : X → Z is a principal T -bundle. All C∗-algebras and Hilbert spaces in this paper will be over C. A continuous-trace algebra A over X is a particularly nice type I C∗-algebra with Hausdorff spectrum X and good local structure (the “Fell condition” [22] — that there are continuously varying rank-one projections in a neighborhood of any point in X). We will always assume A is separable; then a basic structure theorem of Dixmier and Douady [17] says that after stabilization (i.e., tensoring by K, the algebra of compact operators on an infinitedimensional separable Hilbert space H), A becomes locally isomorphic to C0(X,K), the continuous K-valued functions on X vanishing at infinity. However, A need not be globally isomorphic to C0(X,K), even after stabilization. The reason is that a stable continuous-trace algebra is the algebra of sections (vanishing at infinity) of a bundle of algebras over X, with fibers all isomorphic to K. The structure group of the bundle is AutK ∼= PU(H), the projective unitary group U(H)/T. Since U(H) is contractible and the circle group T acts freely on it, PU(H) is an Eilenberg-Mac Lane K(Z, 2)space, and thus bundles of this type are classified by homotopy classes of continuous maps from X to BPU(H), which is a K(Z, 3)-space, or in other words by H3(X,Z). In this way, one can show that the continuous-trace algebras over X, modulo Morita equivalence over X, naturally form a group under the operation of tensor product over C0(X), called the Brauer group Br(X), and that this group is isomorphic to H3(X,Z) via the DixmierDouady class. Given an element δ ∈ H3(X,Z), we denote by CT (X, δ) the associated stable continuous-trace algebra. (If δ = 0, this is simply C0(X,K).) The (complex topological) K-theory K•(CT (X, δ)) is called the twisted K-theory [43, §2] of X with twist δ, denoted K−•(X, δ). When δ is torsion, twisted K-theory had earlier been considered by Karoubi and Donovan [18]. When δ = 0, twisted K-theory reduces to ordinary K-theory (with compact supports). Now recall we are assuming X is equipped with a free T -action with quotient X/T = Z. (This means our theory is “compactified along tori” in a way reflecting a global symmetry group of X.) In general, a group action on X need not lift to an action on CT (X, δ) for any value of δ other than 0, and even when such a lift exists, it is not necessarily essentially unique. So one wants a way of keeping track of what lifts are possible and how unique they are. The equivariant Brauer group defined in [14] consists of equivariant Morita equivalence classes of continuous-trace algebras over X equipped with group actions lifting the action on X. Two group actions on VARGHESE MATHAI AND JONATHAN ROSENBERG 5 the same stable continuous-trace algebra over X define the same element in the equivariant Brauer group if and only if they are outer conjugate. (This implies in particular that the crossed products are isomorphic. However, it is perfectly possible for the crossed products to be isomorphic even if the actions are not outer conjugate.) Now let G be the universal cover of the torus T , a vector group. Then G also acts on X via the quotient map G ։ T (whose kernel N can be identified with the free abelian group π1(T )). In our situation there are three Brauer groups to consider: Br(X) ∼= H3(X,Z), BrT (X), and BrG(X), but BrT (X) is rather uninteresting, as it is naturally isomorphic to Br(Z) [14, §6.2]. Again by [14, §6.2], the natural “forgetful map” (forgetting the T -action) BrT (X) → Br(X) can simply be identified with p∗ : Br(Z) ∼= H3(Z,Z) → H3(X,Z) ∼= Br(X). The basic setup from [33] is Basic Setup ([33, 3.1]). A spacetime X compactified over a torus T will correspond to a space X (locally compact, finite-dimensional, homotopically finite) equipped with a free T -action. Without essential loss of generality, we may as well assume that X is connected. The quotient map p : X → Z is a principal T -bundle. The NS 3-form H on X has an integral cohomology class δ which corresponds to an element of Br(X) ∼= H3(X, Z). A pair (X, δ) will be a candidate for having a T -dual when the T -symmetry of X lifts to an action of the vector group G on CT (X, δ), or in other words, when δ lies in the image of the forgetful map F : BrG(X) → Br(X). Recall from [33] that if T is a torus of dimension n, so that G ∼= R, we have the following facts. Here we denote by H• M (G,A) the cohomology of the topological group G with coefficients in the topological G-module A, as defined in [38]. This is sometimes called “Moore cohomology” or “cohomology with Borel cochains.” We denote by H• Lie(g, A) the Lie algebra cohomology of the Lie algebra g of G with coefficients in a module A. Theorem 1.1 ([33, Theorem 4.4]). Suppose G ∼= R is a vector group and X is a locally compact G-space (satisfying our finiteness assumptions). Then there is an exact sequence: H2(X,Z) d 2 // H2 M (G,C(X,T)) ξ // BrG(X) F // H3(X,Z) 3// H3 M (G,C(X,T)). Lemma 1.2 ([33, equation (5)]). If X is as above, then H M (G,C(X,T)) ∼= H M (G,C(X,R)) for • > 1. Theorem 1.3 ([26, Cor, III.7.5], quoted in [33, Corollary 4.7]). If G is a vector group with Lie algebra g, and if A is a G-module which is a complete 6 T-DUALITY VIA NONCOMMUTATIVE TOPOLOGY, II metrizable topological vector space, then H• cont(G,A) ∼= H• Lie(g, A∞). (Here A∞ is the submodule of smooth vectors for the action of G.) In particular, it vanishes for • > dimG. 2 Main mathematical results Now we are ready to start on the main results. First we begin with a lemma concerning the computation of a certain Moore cohomology group. This lemma is quite similar to [33, Lemma 4.9]. Lemma 2.1. If G is a vector group and X is a G-space (with a full lattice subgroup N acting trivially) as in the Basic Setup above, then the maps p∗ : C(Z,R) → C(X,R) and “averaging along the fibers of p” ∫ : C(X,R) → C(Z,R) (defined by ∫ f(z) = ∫ T f(g · x) dg, where dg is Haar measure on the torus T and we choose x ∈ p−1(z)) induce isomorphisms H M (G,C(X,R)) ⇆ H • M (G,C(Z,R)) ∼= C(Z,R)⊗ ∧•(g∗) which are inverses to one another. Proof. We apply Lemma 1.2 and Theorem 1.3. Note that the G-action on C(Z,R) is trivial, so every element of C(Z,R) is smooth for the action of G. But for any real vector space V with trivial G-action, H M (G,V ) ∼= H Lie(g, V ) ∼= H Lie(g,R)⊗ V ∼= ∧•(g∗)⊗ V. Clearly ∫ ◦ p∗ is the identity on C(Z,R), so we need to show p∗◦ ∫ induces an isomorphism on cohomology of C(X,R). The calculation turns out to be local, so by a Mayer-Vietoris argument we can reduce to the case where p is a trivial bundle, i.e., X = (G/N)×Z, with G acting only on the first factor. The smooth vectors in C(X,R) for the action of G can then be identified with C(Z,C∞(G/N)). So we obtain H M ( G,C(X,R) ) ∼= H Lie ( g, C(Z,C(G/N)) ) ∼= C ( Z,H Lie ( g, C(G/N) )) , with the cohomology moving inside since G acts trivially on Z. However, we have by [3, Ch. VII, §2] that H Lie ( g, C(G/N) ) ∼= H M (N,R) ∼= ∧•(g∗). The first main result generalizes the result in [33] that says that the forgetful map F : BrG(X) → Br(X) is surjective when dimG ≤ 2. In higher dimensions, F need not be surjective, but we characterize its image. VARGHESE MATHAI AND JONATHAN ROSENBERG 7 Theorem 2.2. Let T be a torus, G its universal covering, and p : X → Z be a principal T -bundle, as in the Basic Setup of section 1. Then the image of the forgetful map F : BrG(X) → H 3(X,Z) is precisely the kernel of the map ι∗ : H3(X,Z) → H3(T,Z) induced by the inclusion ι : T →֒ X of a torus fiber into X. Proof. There are two distinct parts to the proof. First we show that if δ ∈ H3(X,Z) and ι∗(δ) 6= 0, then δ cannot be in the image of F . For this (since we can always restrict to a G-invariant set), it suffices to consider the case where X = T with T acting simply transitively. Then if δ lies in the image of F , that means the corresponding principal PU -bundle E → T (with classifying map δ) carries an action of G for which the bundle projection is G-equivariant. This action corresponds to an integrable way of choosing horizontal spaces in the tangent bundle of E, or in other words, to a flat connection on E. That means, since T = G/N with N free abelian, that the bundle, together with itsG-action, comes from a homomorphism ρ : π1(T ) = N → PU , which we can regard as a projective unitary representation of N . In other words, if A = CT (T, δ) carries an action of G compatible with the transitive action of G on T , then the action is induced from the action of N on K associated to a projective unitary representation ρ of N . So we have a surjective “induction” homomorphism H2 M (N,T) ∼= BrN (pt) → BrG(G/N). Consider the Mackey obstructionM(ρ) ∈ H2 M (N,T) ∼= ∧2(N∗)⊗T, which is the class of the N -action on K as an element of BrN (pt). Since H 2 M (N,T) is generated by product cocycles ω1⊗· · ·⊗ωk⊗1, where 2k ≤ n = dimG, N splits as N1×· · ·Nk×N ′ with each Nj of rank 2, and with ωj skew and nondegenerate on Nj , it is enough to show δ is trivial in such a case. But in this case, if we let Gj = Nj ⊗Z R, G ′ = N ′ ⊗Z R, then A = Ind G N (K, ρ) evidently splits as an “external tensor product” Ind1 N1(K, ω1) ⊗ · · · ⊗ Ind Gk Nk (K, ωk) ⊗ Ind ′ N ′(K, 1). Since Ind G N ′(K, 1) ∼= C(G′/N ′)⊗K has trivial Dixmier-Douady class, the Dixmier-Douady class of A is thus determined by the DixmierDouady classes of continuous-trace algebras over the 2-tori Gj/Nj . These of course have to be trivial, since H3(T 2) = 0. This completes the first half of the proof. For the second half of the proof, we have to show that if ι∗δ = 0, then δ is in the image of the forgetful map. We apply Theorem 1.1. This says it suffices to show that δ is in the kernel of the map d3 : H (X,Z) → H M (G,C(X,T)). 8 T-DUALITY VIA NONCOMMUTATIVE TOPOLOGY, II from that theorem. By Lemma 2.1, the calculation of the Moore cohomology group is local (in the base Z), and leads to a natural isomorphism H M (G,C(X,T)) ∼= C(Z,R)⊗H(T,R). Thus d3 must be a map H 3(X,Z) → C(Z,R) ⊗ H3(T,R) which is locally defined (in Z) and natural for all principal T -bundles. It must therefore factor through ι∗, and so everything in the kernel of ι∗ is the image of F . Theorem 2.3. In the Basic Setup with dimG = n arbitrary, there is a commutative diagram of exact sequences: H0(Z, ∧2(Zn)) 0 H2(X,Z) d 2 // H2 M (G,C(X,T)) ξ //