Topological t-duality for twisted tori

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Topological T-Duality for Twisted Tori

Paolo ASCHIERI†1†2†3 and Richard J. SZABO†1†2†4†5†6

†1

Dipartimento di Scienze e Innovazione Tecnologica, Universit`a del Piemonte Orientale, Viale T. Michel 11, 15121 Alessandria, Italy

E-mail: paolo.aschieri@uniupo.it

†2

Arnold–Regge Centre, Via P. Giuria 1, 10125 Torino, Italy

†3

Istituto Nazionale di Fisica Nucleare, Torino, Via P. Giuria 1, 10125 Torino, Italy

†4

Department of Mathematics, Heriot-Watt University,

Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK E-mail: R.J.Szabo@hw.ac.uk

†5

Maxwell Institute for Mathematical Sciences, Edinburgh, UK

†6

Higgs Centre for Theoretical Physics, Edinburgh, UK

Received June 30, 2020, in final form January 22, 2021; Published online February 05, 2021 https://doi.org/10.3842/SIGMA.2021.012

Abstract. We apply the C⇤_{-algebraic formalism of topological T-duality due to Mathai and}

Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative C⇤_{-algebra with an action of}_Rn_{. We treat the general class of}

almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier– Douady classes. We prove that any such solvmanifold has a topological T-dual given by a C⇤ -algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these C⇤-algebras rigorously describe the T-folds from non-geometric string theory.

Key words: noncommutative C⇤-algebraic T-duality; nongeometric backgrounds; Mostow fibration of almost abelian solvmanifolds; C⇤_{-algebra bundles of noncommutative tori}

2020 Mathematics Subject Classification: 46L55; 81T30; 16D90

Dedicated to Giovanni Landi on the occasion of his 60th birthday

1 Introduction

1.1 Background

T-duality is a symmetry of string theory which relates distinct spaces that describe the same physics. It has presented a challenge to mathematics in finding a rigorous framework in which these ‘equivalences’ of spaces is manifest. It was realized early on that noncommutative geometry provides such a framework [18,30], at least in the simplest cases of tori endowed with a trivial gerbe, where subsequently it was shown that T-duality is realised as Morita equivalence of noncommutative tori [5,29,38,47,48].

T-duality of spaces which are compactified on tori, or more generally torus bundles, can be explained topologically in terms of correspondence spaces which implement a smooth analog of the Fourier–Mukai transform [24]. In the correspondence space picture, T-duality transfor-mations are realised as homeomorphisms in the mapping class group of the fibres of doubled torus bundles. This gives rise to an isomorphism of K-theory groups, which are the groups of D-brane charges on the pertinent space; as this only concerns how topological data of the space change under T-duality, it is commonly refered to as ‘topological T-duality’, to distinguish it from the more physical notion of T-duality which also dictates how geometric data on the space should transform. It was shown by [32] that this can be reformulated in terms of the C⇤-algebra of functions on the space by considering its crossed product by an action of the abelian Lie group_Rn_{, leading to a general T-duality formalism that can be regarded as a noncommutative}

version of the topological aspects of the Fourier–Mukai transform; this version of T-duality is often called the ‘C⇤-algebraic formulation’ of topological T-duality.

The story becomes more interesting for spaces that are endowed with a non-trivial gerbe, which in string theory typically comprise torus bundles with ‘H-flux’. The gerbe can be en-coded in the data of a continuous-trace C⇤-algebra with a non-trivial Dixmier–Douady class, which is a noncommutative algebra to which the formalism of topological T-duality was applied originally in [3,32,33]. In addition to relating spaces with di↵erent topologies, T-duality in string theory for such instances predicts the existence of ‘non-geometric’ spaces, called T-folds [25], which cannot be viewed as conventional Hausdor↵ topological spaces. In these instances the

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correspondence space picture ‘geometrizes’ the action of T-duality. It was shown by [4,32,33] that the T-folds of [25] have a rigorous incarnation in noncommutative geometry as C⇤-algebra bundles of noncommutative tori; necessary and sufficient conditions for the existence of ‘classical’ T-dual Hausdor↵ spaces were developed in terms of topological data, and explicit constructions of ‘non-classical’ T-duals as noncommutative torus bundles were given. These points of view were harmonised in [6], and in [7] a C⇤-algebraic version of the correspondence space construc-tion was given. The explicit connecconstruc-tions of these noncommutative torus bundles to the T-folds of [25] in the setting of noncommutative gauge theories on D-branes in T-folds was elucidated in [16, 19, 28, 31]. Topological T-duality and T-folds have also been studied rigorously from other approaches based on homotopy theory [8,9] and on higher geometry [36].

In string theory, the simplest examples of torus bundles are sometimes called ‘twisted to-ri’ [13]; although this name is a misnomer, we continue to use it as it is convenient for our pur-poses. These are fibrations of n-dimensional toriTnover a circleT which do not carry the extra data of a gerbe; they have monodromy in the mapping class group SL(n,Z) of the torus fibers. The simplest examples of these, the Heisenberg nilmanifolds, are T-dual to torus bundles with H-flux and also to T-folds, and they arise in the C⇤-algebraic constructions of [32,33]. However, there are other examples which do not have any classical dual with H-flux, and these are missed by the usual C⇤-algebraic framework which starts from continuous-trace algebras. The simplest examples of these with n = 2 were studied in [28] in the language of noncommutative gauge theo-ries, where it was shown that the monodromy of the original torus bundles becomes a non-trivial Morita equivalence of the fiber noncommutative tori of the dual C⇤-algebra bundle. As far as we are aware, these are new examples of noncommutative torus bundles which have not been rigorously studied in the mathematics literature, and the primary purpose of this paper is to fill this gap: starting from the C⇤-algebra of functions on a twisted torus in any dimension, we give a rigorous construction of the topological T-duals in the C⇤-algebraic framework and precisely describe the non-classical C⇤-algebra bundles with their Morita equivalence monodromies. This includes some of the examples from [32,33] based on topological T-duality applied to continuous-trace algebras, and the examples of [28] based on T-duality in noncommutative gauge theory, while at the same time it produces many new examples. In particular, we give a unified descrip-tion of the noncommutative torus bundles which are T-dual to twisted tori in any dimension.

The noncommutative gauge theory on a D-brane comes with other moduli, in addition to the noncommutativity parameters, which also generally transform in a non-trivial way under the monodromies so as to leave the physics unchanged [28]. In the absence of other moduli, as in topological T-duality, the non-trivial Morita equivalences of the fibres of the C⇤-algebra bundles require an interpretation akin to the topological monodromies, which act as homeomor-phisms in the mapping class group SL(n,Z) of the fibres of the original twisted torus. This is naturally achieved by considering our C⇤-algebras as objects in a category where both the usual ⇤-isomorphisms as well as Morita equivalences are realised as isomorphisms. This category is well-known to experts and all of our considerations of topological T-duality in this paper will take place therein. This perspective will also be advantageous for eventual rigorous consider-ations of noncommutative gauge theories on these C⇤-algebra bundles in terms of projective modules, as well as for the treatments of D-branes in terms of their K-theory, though we do not pursue these further aspects in the present paper.

1.2 Summary and outline

In this paper our starting point is a very general definition of a twisted torus_T⇤G as the quotient of a locally compact group G by a lattice ⇤G in G; this definition encompasses the Tn-bundles

over T discussed above, along with many other known examples from string theory. We re-gard _T⇤G as a ‘torus bundle without H-flux’, which is captured simply by the C⇤-algebra of

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functions C(_T⇤G). This is ultimately the novelty of our approach, which leads to a simpler perspective on topological T-duality as compared to the approach of [32,33] based on the more complicated continuous-trace algebras. Our approach uses similar techniques as those of [32,33] for evaluating Morita equivalences of cross products by actions of_Rn_{, though with a much}

sim-pler C⇤-algebraic structure. In particular, in this paper we do not develop any new C⇤-algebraic machinery as such, but instead we gather a fortuitously existing collection of results that enable us to explicitly identify both classical and non-classical T-duals of twisted tori with relatively straightforward algebraic techniques. On the other hand, the tradeo↵ for the simplicity of our framework is the absence of some key constructions from [4,32,33].

We have endeavoured throughout to provide a fairly self-contained, and at times pedagogical, presentation. For this reason we have collected all the key concepts and tools involving cross products of C⇤-algebras and Morita equivalence in Section 2. Experts versed in C⇤-algebra theory may safely skip this section.

In Section3 we give our definition of twisted tori T⇤G and discuss the C⇤-algebraic formula-tion of topological T-duality that we employ in this paper. We describe how T-dual C⇤-algebras are naturally isomorphic when regarded as objects of the additive category KK that underlies Kasparov’s bivariant K-theory, and we adapt the construction of noncommuative correspon-dences from [7] as diagrams in this category. We spell out some simple techniques that we use to compute classical T-duals with H-flux, i.e., the T-dual algebra is a certain continuous-trace C⇤ -algebra with non-trivial Dixmier–Douady class, and more general techniques based on Green’s symmetric imprimitivity theorem which enable the computation of noncommutative T-duals. We illustrate our scheme on two well-known examples which have classical T-duals: we repro-duce the standard rules for T-duality of tori as well as the topology changing mechanism for T-duality of orbifolds of compact Lie groups G.

In Section4we come to the main class of examples and results of this paper. We review the definition and topology of the special class of twisted tori provided by almost abelian solvman-ifolds, which are Tn-bundles over a circle T. Accordingly, we regard the algebra of functions C(_T⇤G) as an object in the category RKKT of C⇤-algebra bundles over T, where in particular fibrewise Morita equivalences are isomorphisms. T-duality in this category requires fibrewise actions ofRn, and in particularRn-actions which act non-trivially on the baseT would take the algebra out of the category _RKK_T. This means that the ‘essentially doubled spaces’ of [28], which arise from T-duality along the base circle and require a completely doubled formalism, are not considered in this paper; they require working in a di↵erent category, which we do not discuss here. We give necessary and sufficient criteria for the existence of classical T-duals with H-flux in this case which are based on simple algebraic data of the underlying group G, and we explicitly compute the corresponding continuous-trace C⇤-algebras dual to any almost abelian solvmanifoldT⇤G satisfying these conditions. We further show that any such solvmanifold has a non-classical T-dual that is a C⇤-algebra bundle of noncommutative n-tori over T, which we also compute explicitly; this rigorously confirms, in particular, arguments from string theory suggesting that non-geometric solutions result from T-duality on some six-dimensional almost abelian solvmanifolds [1].

Finally, Section 5 is devoted to explicit examples of the general formalism of Section4. We apply our results to all three classes of three-dimensional solvmanifolds. We recover in this way a new perspective on the well-known T-duals of the Heisenberg nilmanifolds: the three-torusT3 with H-flux, and the basic noncommutative principalT2-bundle overT given by the group C⇤ -algebra of the integer Heisenberg group. Our general formalism also rigorously reproduces the noncommutative torus bundles from [28] which are T-dual to Euclidean solvmanifolds for the_Z4

elliptic conjugacy class of SL(2,Z). We extend these results to give new examples of noncommu-tative torus bundles dual to Euclidean solvmanifolds for theZ2 andZ6 elliptic conjugacy classes,

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formulation of topological T-duality to the case of non-principal torus bundles, which have also been previously considered in [22].

2 Crossed products and duality

In this section we summarise some of the mathematical tools that we will use in this paper. A good reference for the material covered in the following is the book [52]. Throughout this paper, all topological spaces are assumed to be second countable (hence separable), locally compact and Hausdor↵.

2.1 Dynamical systems and their crossed products

Let G be a locally compact group and let X be a G-space, i.e., a topological space which is acted upon homeomorphically by G; we denote the G-action G_{⇥ X ! X by ( , x) 7! · x. The} pair (X, G) is called a transformation group. A related concept is that of a dynamical system, which is a triple (_{A, G, ↵) consisting of an algebra A and a locally compact group G acting} on _{A via a group homomorphism ↵: G ! Aut(A), denoted} _{7! (↵ : A ! A) for} _{2 G. In} topological T-duality one usually requiresA to be a C⇤_{-algebra, in which case (}_{A, G, ↵) is called}

a C⇤-dynamical system. Two dynamical systems (_{A, G, ↵) and (B, G, ) are equivalent if there} is an algebra isomorphism ' :_{A ! B which intertwines the G-actions: ' ↵ =} ' for all

2 G.

If A is a commutative C⇤_{-algebra, then we call (}_{A, G, ↵) a commutative dynamical system.}

In that case, by Gelfand duality A = C0(X) is the algebra of continuous functions

vanish-ing at infinity on a topological space X equipped with the G-action ↵† _X for _{2 G under} the identification of points x 2 X with irreducible representations of C0(X), which are

one-dimensional and given by the point evaluation maps evx: A ! A with evx(f) = f(x); then

(X, G) is a transformation group. Conversely, given a transformation group (X, G), there is an associated commutative dynamical system (C0(X), G, ↵), where ↵ (f)(x) = f 1 · x for

2 G, f 2 C0(X) and x 2 X. In other words, there is a one-to-one correspondence between

transformation groups and commutative C⇤-dynamical systems.

If the C⇤-algebra _{A is not commutative, then we call (A, G, ↵) a noncommutative C}⇤ -dynamical system.

As usual, it is more useful to work with a representation rather than the abstract dynamical system itself. A covariant representation of a dynamical system (A, G, ↵) in a C⇤_-algebra _B

with multiplier algebra M(_{B) is a pair (⇧, U) consisting of a homomorphism ⇧ : A ! M(B) and} a unitary representation U : G_{! M(B),} _{7! U which satisfy the compatibility condition}

⇧ ↵ (a) = U ⇧(a)U 1,

for all 2 G and a 2 A. A natural choice is to take B = K(H) to be the C⇤_{-algebra of compact}

operators on a separable Hilbert space _{H, which gives a representation ⇧ : A ! B(H) of the} algebra_{A by bounded operators B(H) on H and a unitary representation U : G ! B(H) of the} group G onH; in this case we call (⇧, U) a covariant representation of (A, G, ↵) on H.

When a group G acts on a space X, one is naturally interested in considering the quotient space X/G of G-orbits on X. When G acts freely and properly on X, this is described alge-braically by the algebra of functions C0(X/G). More generally, the subalgebra of G-invariant

elements _AG _{✓ A of a G-algebra A can be used to represent the quotient, even for G-actions}

with fixed points. A more general and systematic way of dealing with the e↵ective algebraic “quotient” is through the crossed product algebra_{A o}↵G for a dynamical system (A, G, ↵). For

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quotient X/G is not a Hausdor↵ space, while for a free and proper G-action it gives an algebra with the same spectrum X/G as the algebra C0(X/G) = C0(X)G of G-invariant functions on X.

In order to define the crossed product algebra, we first define kfkuniv:= sup

(⇧,U )

(⇧o↵U )(f )

for compactly supported functions f 2 Cc(G,A), where the supremum is taken over (possibly

degenerate) covariant representations (⇧, U ) of (_{A, G, ↵) with} (⇧o↵U )(f ) :=

Z

G

⇧ f ( ) U dµG( ),

and µG denotes the left invariant Haar measure on G. This defines a norm, called the universal

norm, on the space Cc(G,A). Then the crossed product algebra A o↵G is the completion (in

the universal norm) of the algebra Cc(G,A) equipped with the convolution product

(f ? f0)( ) := Z

G

f ( 0)↵ 0 f0 0 1 dµG( 0), (2.1)

for all f, f0: G! A. In general this is a noncommutative multiplication, even for commutative dynamical systems. Since _{A is a C}⇤-algebra, there is a _{⇤-structure on the convolution algebra} defined by

f†( ) := G( ) 1↵ f 1 ⇤ ,

where G: G! R+ is the modular function of G defined through G( 0) Z G f ( ) dµG( ) = Z G f ( 0) dµG( )

for f 2 Cc(G,A) and 0 2 G. By the uniqueness of the left invariant Haar measure µG up to

a positive constant, G( 0) is independent of f and G is easily proven to be a continuous group

homomorphism from G to the multiplicative group_R+_{; it is trivial for abelian groups and for}

compact groups.

When_{A = C}c(X) is the algebra of a commutative dynamical system, the convolution algebra

Cc(G⇥ X) consists of functions f : G ⇥ X ! C and the convolution product reads as

(f ? f0)( , x) = Z

G

f ( 0, x)f0 0 1 , 0 1· x dµG( 0),

while the_{⇤-algebra structure is given by} f†( , x) = G( ) 1f 1, 1· x .

The crossed product is a generalization of the usual group algebra C⇤(G), the completion (in the universal norm) of Cc(G) which is recovered in the caseA = C (the C⇤-algebra of a point) wherein

↵ = id_A:A ! A for all 2 G and (2.1) recovers the usual convolution product of functions on the group G. As explained in Section2.2below, the group C⇤-algebra description illustrates the relation between crossed products and semi-direct products of groups (see Theorem 2.5). We also note that if a group G acts trivially on an algebraA, then A o G ' A ⌦ C⇤_(G).

The crossed product can be thought of as a universal object for covariant representations of the dynamical system (_{A, G, ↵), in the following sense: Define the universal covariant} represen-tation (⇧, U ) of (A, G, ↵) in A o↵G by

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for a _{2 A, f 2 C}c(G,A) and , 0 2 G. Then the universal property defining the crossed

product implies that any covariant representation (⇧, U ) of (A, G, ↵) in a C⇤_-algebra _B

fac-tors through the universal covariant representation: There exists a unique homomorphism ' : M(_{A o}↵G)! M(B) such that

⇧ = ' ⇧ and U = '(U )

for all 2 G.

If (⇧, U ) is a covariant representation of the dynamical system (_{A, G, ↵) on a Hilbert space H,} then

(⇧,U )(f ) := (⇧o↵U )(f )

defines a representation (⇧,U ): Cc(G,A) ! B(H) of the crossed product A o↵G as bounded

operators on _{H. This is called the integrated form of the covariant representation (⇧, U). In} particular, it maps the convolution product (2.1) onto the operator product in the algebra B(_H),

(⇧,U )(f ? g) = (⇧,U )(f ) (⇧,U )(g),

and it is covariant in the sense that

(⇧,U ) iG( )(f ) = U (⇧,U )(f ),

where iG( )(f ) ( 0) := ↵ f 1 0 for each , 0 2 G and f 2 Cc(G,A).

Example 2.1 (noncommutative two-tori). The noncommutative torus is a fundamental example of a noncommutative space in both physics and mathematics. Its original incarnation [43] is a nice example of a crossed product construction, which will play a fundamental role later on in this paper. Let C(T), Z, ⌧✓ _{be the commutative C}⇤_{-dynamical system where ⌧}✓ _{is induced}

through pullback by rotations of the circleT through a fixed angle ✓ 2 R/Z: ⌧_n✓(f)(z) = f e2⇡in✓z ,

for n2 Z, f 2 C(T) and z 2 T. The resulting crossed product A✓ := C(T) o⌧✓Z

is a called a rotation algebra, and for irrational values of ✓ it can be identified as a noncommu-tative two-torus_T2

✓ in the following way.

By definition, the algebra A✓ is the universal norm completion of the convolution algebra

Cc(Z ⇥ T), whose elements f = {fn}n2Z can be regarded as sequences (with only finitely many

nonvanishing terms) of functions fn:T ! C. The convolution product is given by

(f ?✓g)n(z) :=

X

n0_2Z

fn0(z)gn n0 e2⇡in0✓z ,

and the_{⇤-algebra structure is} f_n†(z) := f n e2⇡in✓z .

Via the Fourier transformation f (z, w) :=X

n2Z

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for w _{2 T, we may regard the convolution algebra C}c(Z ⇥ T) as a subspace of the space of

functions C T2 equipped with the star-product (f ?✓g)(z, w) =

X

n2Z

(f ?✓g)n(z)wn. (2.2)

After a further Fourier transformation fn(z) =

X

m2Z

fm,nzm

and some simple redefinitions of the Fourier series involved, the star-product (2.2) may be written in the form

(f ?✓g)(z, w) = X (m,n)2Z2 ✓ _X (m0,n0)2Z2 fm0_,n0g_{m m}0_{,n n}0e2⇡i(m m0)n0✓ ◆ zmwn.

This recovers the usual commutative pointwise multiplication of functions in C T2 _{for ✓ = 0.}

For ✓ 6= 0 it realizes the irrational rotation algebra A✓ as a deformation of the algebra of

functions C T2 on a two-dimensional torus T2; it is equivalent to the usual strict deformation quantization ofT2 _{whose star-product is a twisted convolution product on C} _T2 _.

In the language of covariant representations of the dynamical system (C(T), Z, ⌧✓), the crossed product A✓ is the universal C⇤-algebra generated by two unitaries U and V satisfying the

rela-tion [52, Proposition 2.56]

U V = e 2⇡i✓V U. (2.3)

A concrete representation of A✓ on the Hilbert spaceH = L2(T) is given by defining

U (f)(z) = zf(z) and V (f)(z) = f e2⇡i✓z .

Example 2.2 (Noncommutative d-tori). The natural higher-dimensional generalization of Example 2.1 involves a skew-symmetric real d⇥d matrix ⇥ = (✓ij), see [45]. Then the

non-commutative d-torus A⇥ =Td⇥ is the universal C⇤-algebra generated by d unitaries U1, . . . , Ud

satisfying the relations UiUj = e 2⇡i✓ijUjUi

for i, j = 1, . . . , d. By [37, Lemma 1.5], every noncommutative torus Td_⇥ can be obtained as an iterated crossed product by _{Z in the following way. Let ⇥}_{|d 1} = (✓ij)1i,jd 1, and let

U1, . . . , Ud 1 be the standard generators of A⇥_{|d 1} = Td 1_⇥_{|d 1}. Define a group homomorphism

⌧~✓_:_{Z ! Aut(A}

⇥_{|d 1}) by

⌧_n✓~(Ui) = e2⇡in✓idUi,

for n2 Z, where ~✓ := (✓id)2 Rd 1. Then there is an isomorphism of C⇤-algebras

A⇥ ' A⇥_{|d 1}o_⌧~✓Z. (2.4)

In the particular case where ⇥_{|d 1} = 0d 1, we denote the corresponding noncommutative

d-torus by A_~_✓ = _Td ~

✓, and (2.4) shows that it can be obtained by a crossed product of the

commutative algebra of functions on a d 1-torus by an action ofZ: A~_✓ ' C Td 1 o_⌧~✓ Z.

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2.2 Semi-direct products and group algebras

Most of our considerations later on will focus on spaces that can be obtained from semi-direct products of groups. We will now explain the relation between crossed products and semi-direct products which will be useful for these examples.

There are two ways to think about the semi-direct product construction:

(1) Let G be a group with two subgroups N and H such that N is normal. If N_{\ H = {e} ⇢ G} and every element of G can be written as a product of an element of N with an element of H, then we say that G is a semi-direct product of its subgroups N and H and we write G = N H.

(2) Let N and H be two groups together with a left action ' : H _{! Aut(N) of H on N by} automorphisms, which we denote by 'h(n) = hn for h 2 H and n 2 N; in particular h_(nn0_{) =}h_nh_n0_{. We write}H_{N to indicate that H acts on N from the left. The semi-direct}

product of N and H is the group No'H defined to be the set N⇥ H with the product

(n, h) (n0, h0) = nhn0, hh0 .

The inverse is then (n, h) 1 = h 1n 1, h 1 .

These two definitions are equivalent: Given subgroups N, H_{⇢ G as in point (1), it follows that} G ' N oAdH where Ad is the adjoint action: Adh(n) = hnh 1. On the other hand, every

element of the group G = No'H defined in (2) can be written as (n, h) = (n, eH)(eN, h) and the

subgroups N_{⇥ {e}H} and {eN} ⇥ H intersect only in the identity of G. If the action of H on N is

trivial, i.e., 'h = idN for all h2 H, then the semi-direct product reduces to the direct product

No'H = N⇥ H.

Later on we will need to consider the interplay between semi-direct products and quotient groups, which is provided by the simple

Lemma 2.3. Let G = N_o'H be a semi-direct product, and let V⇢ N be a subgroup which is

normal in G. Then the quotient group G/V is the semi-direct product (N/V)o_'VH, where 'V is the action ' of H induced on the quotient group N/V.

If N is a group, we write C⇤(N) for the corresponding group C⇤-algebra, i.e., for the crossed product C o N. If N is finite, then C⇤(N) = C[N] is the linear space freely generated over C by the group elements, made into an algebra by linearly extending the product from N toC[N]; equivalently it is the algebra of continuous functions on N with the convolution product. Given a left H-action ' : H _{! Aut(N), there is an induced action '}⇤_{: H}_{! Aut(C}⇤_{(N)) via pullback.}

For H and N finite the vector spaces C⇤(N)⇥ H and C⇤_(N_{⇥ H) are canonically isomorphic, and}

it is straightforward to show that the corresponding crossed product and semi-direct product are related by

Proposition 2.4. If N and H are finite groups, then C⇤(N)o'⇤H' C⇤(No'H).

Proof . Note that C⇤(N)_o'⇤H =C[N] o'⇤H is the vector space of functions f : H! C[N] with convolution product

f ?_C[N]o_'⇤_Hf0 (h) = X

h0_2H

f (h0) ?_C[N]'⇤_h0 f0 h0 1h ,

and using the convolution product inC[N] this can be written as f ?_C[N]o_'⇤Hf0 (n, h) = X h0_2H X n0_2N f (n0, h0)f0 h0 1 n0 1n , h0 1h ,

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A more general result holds if N and H are locally compact groups with ' : H _{! Aut(N)} a continuous action of H on N via automorphisms (i.e., (h, n)7! 'h(n) is a continuous map from

H⇥ N to N). In this case the semi-direct product N o'H is a locally compact group (in the

product topology on N_{⇥ H) with N a closed normal subgroup and H a closed subgroup (see [}52, Proposition 3.11] with A = C). The analogue of item (1) above also holds in the context of locally compact groups if G is -compact, and N and H are closed subgroups of G.

The action defining the C⇤-dynamical system (C⇤(N), H, ) and hence the crossed product C⇤(N)o H is the composition of the pullback '⇤ of the action ' : H! Aut(N) with the action

H: H! R+⇢ Aut(C⇤(N)) that enters the definition of the Haar measure on No'H in terms

of the Haar measures on N and H: If µN is a (left invariant) Haar measure on N, then the

integral Ih(F ) =

R

NF hn dµN(n) for F 2 C⇤(N) is left invariant, i.e., Ih( n0F ) = Ih(F ) where

( n0F )(n) := F n0 1n for all n0 2 N (use invariance of the Haar measure under n ! h 1n0). Uniqueness of the Haar measure up to a positive constant then implies there exists a function

H: H! R+ such that H(h) Z N F hn dµN(n) = Z N F (n) dµN(n). (2.5)

It is straighforward to see that H is a group homomorphism and that it is continuous [52,

Section 2]. The Haar measure µN_o'H on No'H is then given by Z No'H f (n, h) dµNo'H(n, h) := Z H Z N f (n, h) H(h) 1dµN(n) dµH(h).

This is trivially invariant under the left N-action, and it is also invariant under the left H-action (n, h)_{7! (1, h}0)(n, h) = h0_{n, h}0_{h , using (}_2.5_{) with F} h0_{n := f} h0_{n, h}0_{h and recalling that h is}

fixed in (2.5).

Theorem 2.5. Let N and H be locally compact groups and ' : H_{! Aut(N) a continuous action} of H on N. Define : H ! Aut(C⇤_{(N)) by (}

h0`)(n) = _H(h0) 1` h0 1n for all h0 2 H and `_{2 C}c(N). Then

C⇤(N)_{o H ' C}⇤(N_o'H).

For a full proof of Theorem 2.5 that takes into account the topological and C⇤-algebraic aspects, see [52, Proposition 3.11]. Here we shall just show that under the canonical injection Cc(No'H) ,! Cc(N)o H, given by f(n, h) 7! f(h) where f(h)(n) = f(n, h), the convolution

product in Cc(N o' H) is mapped to the convolution product in Cc(N)o H. Let f, f0 2

Cc(No'H), then f ?_C_c_(N_o_'_H)f0 (n, h) = Z H Z N f (n0, h0)f0 (n0, h0) 1(n, h) H(h0) 1dµN(n0) dµH(h0). (2.6)

On the other hand, for the images of f , f0 in Cc(N)o H we have

f ?Cc(N)o Hf0 (h) = Z

H

f (h0) ?C⇤(N) h0 f0 h0 1h dµ_H(h0).

Using the convolution product in C⇤(N) this can be written as

f ?_C_c_(N)_{o H}f0 (h)(n) = Z H Z N f (h0)(n0) h0 f0 h0 1h n0 1n dµN(n0) dµH(h0),

which from the definition of the action is easily seen to equal the image (f ?Cc(No'H)f0)(h)(n) in Cc(N)o H of the product (f ?Cc(No'H)f0)(n, h) in Cc(No'H) from (2.6).

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Theorem 2.6. Let (_{A, N o}'H, ↵) be a C⇤-dynamical system for the semi-direct product group

No'H. Then (A o↵|NN, H, ) is a C⇤-dynamical system, where : H _{! Aut(A o}_↵_|_NN), h_{7 !} h

is defined by h(f ) (n) = H(h) 1↵h f h 1

n for all f 2 Cc(N,A) ⇢ A o↵|NN, with H: H! R+ _{defined by (}_2.5_{) and} h 1

n = '_h 1(n). Moreover, the canonical injection C_c(No_'H,A) ,! Cc H, Cc(N,A) extends to a C⇤-algebra isomorphism

A o↵ No'H ' A o↵|NN o H.

Theorem2.5is then recovered by setting _{A = C.}

In the spirit of Theorem 2.5, which shows that crossed products are a generalization of semi-direct products, let us mention the semi-direct product construction behind Theorem2.6. Consider three groups M, N and H with group actionsHN andNoHM; then there are also group actions NM and HM. The associativity of the triple semi-direct product construction is then easily established through

Proposition 2.7. Let M, N and H be groups with group actionsHM, H_{N and}N_{M satisfying the}

compatibility conditions

h n_{m =}(h_{n) h} m

for all m_{2 M, n 2 N and h 2 H. Then there exists a group action} NoH_{M defined by} (n,h)_{m =} n h_{m , and a group action}H_(M_{o N) defined by} h_{(m, n) =} h_m,h_{n , which together satisfy the}

associativity property

Mo (N o H) = (M o N) o H.

2.3 Pontryagin duality and Fourier transform

If N is a locally compact abelian group we denote by bN its Pontryagin dual, i.e., the set of characters : N_{! U(1), which is also a locally compact abelian group (with the compact-open} topology and with the pointwise multiplication). For example, if N =Rd then bN =Rd and the characters are given by p(x) = e2⇡ihp,xi for x2 N and p 2 bN. The Pontryagin duality theorem

states that there is a canonical isomorphism bNb ' N, where n 2 N is associated to the character 7! (n) on bN.

The Fourier transform shows that the group C⇤-algebra C⇤(N) is isomorphic to C0 N : Givenb

a Haar measure µN on N, the Fourier transform F(f) of f 2 Cc(N) is defined by

F(f)( ) := Z

N

f (n) (n) dµN(n)

for _{2 b}N. It sends the convolution product of functions in C⇤(N) to the pointwise product of functions in C(bN):

F(f ? f0) =_F(f)F(f0),

and extends to an isomorphism [52, Proposition 3.1]

F : C⇤(N) '_{! C}0 N ,b (2.7)

where C0 N is the algebra of functions on bb N vanishing at infinity. For N separable, Hausdor↵

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Given a continuous left group action ' : H _{! Aut(N), which we also denote as before by} 'h(n) =hn, consider the induced action : H! Aut C⇤(N) as defined in Theorem2.5. There

is also an induced left action ' : H_b _{! Aut b}N defined by pullback: 'bh (n) := h 1

n , together with its pullback '_b⇤: H _{! Aut C}0 Nb defined by 'b_h⇤f ( ) = bb f 'bh 1 for all h 2 H, bf 2 C0 N andb 2 bN. The Fourier transform isomorphism (2.7) then extends to the

isomorphism

Proposition 2.8. If N is a locally compact abelian group and ' : H_{! Aut(N) is a continuous} action of a locally compact group H on N, then

C⇤(N)_{o H ' C}0 Nb o'b⇤H.

Proof . We show that the triples (C⇤(N), H, ) and C0 N , H,b 'b⇤ are equivalent dynamical

systems, see [52, Example 3.16]. For this, we prove that the Fourier transform (2.7) is H-equivariant with respect to the H-actions and '_b⇤. For h _{2 H, f 2 C}c(N) and 2 bN we

compute F h(f ) ( ) = Z N h (f )(n) (n) dµN(n) = H(h) 1 Z N f h 1n (n) dµN(n) = H(h) 1 Z N f h 1n h h 1n dµN(n) = Z N f (n) hn dµN(n) =F(f) b'_h 1 = 'b_h⇤F(f) ( ),

where in the fourth equality we used (2.5) with F h 1n = f h 1n h h 1n . ⌅ Replacing N with bN in Proposition2.8we also obtain the isomorphism

C⇤ Nb _o_bH' C0 bbN o_bb'⇤H' C0(N)o'⇤H, (2.8)

where b: H_{! Aut C}⇤(bN) is defined by bh( bf )( ) = bH(h) 1fb'bh 1 for h 2 H, bf 2 C_c(bN) and 2 bN, with bH: H! R+ defined as in (2.5) but using the dual group bN instead of N, and

in the final isomorphism we used Pontryagin duality bNb ' N.

Another important property of crossed products is Takai duality [52, Section 7.1]. If G is a locally compact abelian group and (A, G, ↵) is a C⇤_{-dynamical system, then (}_{A o}

↵G, bG,↵) isb

a C⇤-dynamical system, where

b

↵ : bG ! Aut(A o↵G), 7 ! b↵

is defined by↵ (f )( ) := ( ) f ( ) for all f_b _{2 C}c(G,A).

Theorem 2.9 (Takai duality). Let (_{A, G, ↵) be a C}⇤-dynamical system where G is a locally compact abelian group. Then there is an isomorphism of C⇤-algebras

(A o↵G)o↵_bGb' A ⌦ K L2(G) .

2.4 Morita equivalence and Green’s theorem

Crossed products of algebras provide a host of examples of dualities which come in the form of various levels of strong and weak equivalences of algebras, see, e.g., [6]. The most primitive form of such dualities is provided by (strong) Morita equivalence [41]. A bimodule for a pair of algebras_{A and B is a vector space M which is simultaneously a left A-module and a right} B-module, where the left action of A commutes with the right action of B: (a · ⇠) · b = a · (⇠ · b)

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for all a_{2 A, b 2 B and ⇠ 2 M. If A and B are C}⇤-algebras, we say that a bimodule_{M is an} A–B Morita equivalence bimodule (or imprimitivity bimodule) if it is equipped with an A-valued inner product_A_{h · , · i and a B-valued inner product h · , · i}_B satisfying the associativity condition

Ah , i · ⇠ = · h , ⇠iB,

for all , , ⇠ 2 M, under which M is complete in the norm closures, and such that the ideal

AhM, Mi is dense in A and hM, MiB is dense in B. The bimodule M establishes a Morita

equivalence between the algebras_{A and B, and in this case we write A ⇠}MB.

Morita equivalent C⇤-algebras have equivalent categories of nondegenerate⇤-representations: If ⇧_B:_{B ! B(H}_B) is a representation of_{B on a Hilbert space H}_B, then we can construct another Hilbert space

HA :=M ⌦BHB

which is the quotient of the tensor product_{M ⌦ H}_B by the relation (⇠_{· b) ⌦ ⇠ ⌦ ⇧}_B(b) = 0 identifying theB-actions for ⇠ 2 M, b 2 B and 2 HB. The inner product onHA is given by

⌦

⇠⌦B ⇠0⌦B 0↵_H_A :=⌦ ⇧B h⇠, ⇠0iB 0↵_H_B,

and a representation ⇧_A:_{A ! B(H}_A) of the algebra_{A is defined by} ⇧_A(a)(⇠⌦B ) = (a· ⇠) ⌦B

for a_{2 A and ⇠ ⌦}_B _{2 H}_A; this representation is unitary equivalent to the representation ⇧_B. Conversely, starting with a representation of_{A, we can use a conjugate B–A equivalence} bimod-uleM to construct a unitary equivalent representation of B; then there are surjective bimodule homomorphisms M ⌦BM ! A and M ⌦AM ! B which satisfy a certain transitivity law.

As a particular consequence of this equivalence, Morita equivalent algebras have homeomorphic spectra and isomorphic K-theory groups.

Example 2.10 (noncommutative two-tori). A famous example of Morita equivalence in both mathematics and string theory is provided by the noncommutative tori A✓ = T2✓ from

Exam-ple 2.1. Firstly, notice from (2.3) that changing the coset representative ✓ _{2 R/Z yields an} identical algebra: A✓+m = A✓ for all m 2 Z. Secondly, there is an obvious C⇤-algebra

isomor-phism A ✓ ' A✓ obtained by interchanging the two generators U and V . The converse is also

true [43, 44]: A✓0 ' A✓ if and only if ✓0 = ✓ mod 1. More generally, two irrational rotation

C⇤-algebras A✓ and A✓0 are Morita equivalent if and only if ✓ and ✓0 lie in the same orbit under the action of GL(2,Z) by fractional linear transformations

✓0= M[✓] := a✓ + b c✓ + d for M= ✓ a b c d ◆ 2 GL(2, Z).

The explicit Morita equivalence bimodules can be found in [43]. On the other hand, the rational rotation algebras A✓ are all Morita equivalent to the commutative algebra C T2 of functions

on the two-torus [44].

Example 2.11 (noncommutative d-tori). The Morita equivalences of Example2.10 generalize to the higher-dimensional noncommutative tori A⇥ = Td⇥ from Example 2.2 in the following

way [46]. First of all, the algebra A⇥ is unchanged if the matrix ⇥ is written in another basis

ofZd: if B2 GL(d, Z) with transpose Bt_{, then there is a C}⇤_{-algebra isomorphism A}

Bt_{⇥ B}' A_⇥. More generally, consider the set of real skew-symmetric d_{⇥d matrices ⇥ whose orbits M[⇥] are} defined for all M _{2 SO(d, d; Z), where}

M [⇥] = (A ⇥ + B) (C ⇥ + D) 1 for M = ✓ A B C D ◆ 2 SO(d, d; Z),

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and A, B, C and D are d_{⇥d block matrices satisfying}

AtC + CtA = 0 = BtD + DtB and AtD + CtB = d.

The set of all such matrices is dense in the space of all skew-symmetric real d_{⇥d matrices, and} there is a Morita equivalence

AM [⇥] ⇠M A⇥.

However, for d > 2 the converse is not generally true: In fact, there are algebras A⇥ and A⇥0 that are isomorphic (and so Morita equivalent) but for which the matrices ⇥ and ⇥0 do not belong to the same SO(d, d;Z) orbit [46].

We will also need an equivariant version of Morita equivalence in order to show that Morita equivalent algebras induce Morita equivalent crossed products according to [14, Section 5.4] Theorem 2.12. Let (A, G, ↵) and (B, G, ) be C⇤_{-dynamical systems such that} _{A and B are}

Morita equivalent. Then the crossed product C⇤-algebras_Ao↵G andBo G are Morita equivalent

if there exists a G-equivariant A–B Morita equivalence bimodule M, i.e., if there is a strongly continuous action U : G! Aut(M) of G on an A–B Morita equivalence bimodule M such that

U (a· ⇠) = ↵ (a) · U (⇠) and U (⇠· b) = U (⇠) · (b), and

AhU (⇠), U (⇠0)i = ↵ Ah⇠, ⇠0i and hU (⇠), U (⇠0)iB = h⇠, ⇠0iB ,

for all _{2 G, ⇠, ⇠}0 _{2 M, a 2 A and b 2 B.}

In this paper, our main application of Morita equivalence will involve Green’s symmetric imprimitivity theorem. Let X be a locally compact space, and let H and K be locally compact groups with commuting free and proper actions on the right and on the left on X, respectively. We can lift these actions to left actions on C0(X) by defining (hf)(x) = f h 1·x and kf (x) =

f(x· k) for all f 2 C0(X), x2 X, h 2 H and k 2 K. Commutativity of the actions of H and K

implies that there are well-defined induced actions of H and K respectively on the quotient spaces K_{\X and X/H, and hence respectively on the algebras C}0(K\X) and C0(X/H) which we

denote rt and lt. Green’s symmetric imprimitivity theorem then reads as [52, Corollary 4.10] Theorem 2.13. There is a Morita equivalence of C⇤-algebras

C0(K\X) ortH⇠M C0(X/H)oltK (2.9)

implemented by the Morita equivalence (or imprimitivity) bimodule M which is the completion of Cc(X) with the actions

(a_{· ⇠)(x) =} Z K a(k, x_{· H)⇠ k} 1_{· x} K(k)1/2dµK(k), (⇠· b)(x) = Z H ⇠ x· h 1 b h, K· x · h 1 H(h) 1/2dµH(h),

for all x2 X, a 2 Cc(K⇥ X/H), b 2 Cc(H⇥ K\X) and ⇠ 2 Cc(X), and the inner products Ah⇠, ⇠0i(k, x · H) = K(k) 1/2 Z H ⇠(x_{· h)⇠}0 _k 1_{· x · h dµ} H(h), h⇠, ⇠0iB(h, K· x) = H(h) 1/2 Z K ⇠ k 1_{· x ⇠}0 _k 1_{· x · h dµ} K(k),

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Theorem 2.13 has several useful applications and corollaries, see, e.g., [42]. A particularly relevant special case that we shall use below is when K is the trivial group, in which case (2.9) reduces to the Morita equivalence

C0(X)ortH⇠M C0(X/H),

illustrating the use of crossed products in describing quotients. This equivalence can in fact be strengthened to a stable isomorphism [42]

C0(X)ortH' C0(X/H)⌦ K L2(H) , (2.10)

whereK denotes the algebra of compact operators.

Example 2.14 (tori). A particularly relevant example for us is the case X =_Rd _{with H =}_Zd

acting by translations (n, x)7! x + n for n 2 Zd _{and x}_{2 R}d_{, which realizes the d-dimensional}

torusTd=Rd/Zdas a crossed product: C Td _⇠M C0 Rd ortZd.

Let us illustrate the construction explicitly. The convolution algebra Cc Zd⇥Rd ⇢C0 Rd ortZd

can be identified with the space of sequences f ={fn}_n2Zd of functions fn:Rd ! C with the

convolution product (f ? g)n(x) =

X

m2Zd

fm(x)gn m(x m).

Consider the algebra

A := f 2 C0 Rd,K `2 Zd f (x + m) = Umf (x)Um1 ,

where Um is the unitary shift operator on `2 Zd defined by (Uma)n = an m for each m 2 Zd

and a ={an}_n2Zd. Define a map : Cc Zd⇥ Rd ! Cc Rd,K `2 Zd by

(f )(x) _mn= fm+n(x + n).

It is easy to see that (f )(x+m) = Um (f )(x)Um1for all f 2 Cc Zd⇥Rd , and if a = (amn)2 A

then defining fn := a0n gives (f ) = a. It is also easy to check that (f ? g) = (f ) (g), and

consequently gives an algebra isomorphism : C0 Rd ortZd '! A.

The explicit Morita equivalence bimodule is now obtained from the completion of M = ⇠ 2 Cc Rd, `2 Zd ⇠(x + m) = Um⇠(x) .

The left action of the algebraA = Cc Zd⇥ Rd is by left matrix multiplication onM:

A ⇥ M ! M,

(a, ⇠)_{7 ! a · ⇠,} (a_{· ⇠)}n=

X

m2Zd

anm⇠m,

while the right action of the algebra C _Td _{is by right pointwise multiplication on} _M:

M ⇥ C Td ! M,

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The left and right inner products are respectively given by Ah⇠, ⌘i = ⇠ ⌦ ⌘⇤, h⇠, ⌘iC(Td₎ = X n2Zd ⇠n⌘n,

for ⇠, ⌘2 M. Together with the isomorphism , this establishes a Morita equivalence between the algebras C0 Rd ortZd and C Td .

Another important class of examples is provided by taking X = G to be a locally compact group with closed subgroups K and H acting respectively by left and right multiplication on G. In particular, in the special case K = G, so that C0(K\G) = C, Theorem 2.13 gives a Morita

equivalence between the commutative dynamical system (C0(G/H), G, lt), with K = G acting

by left multiplication on the homogeneous space G/H (so that (lt f )(x) = f 1x for all f 2 C0(G/H), x2 G/H and 2 G), and the group C⇤-algebra C⇤(H) =C o H:

C0(G/H)oltG⇠M C⇤(H).

This equivalence also follows from the C⇤-algebra isomorphism [52, Theorem 4.29]

C0(G/H)oltG' C⇤(H)⌦ K L2(G/H) .

3 Topological T-duality and twisted tori

In this section we shall apply the results of Section 2, and in particular Green’s theorem, to present a scheme that will be employed in our study of topological T-duality. We shall then illustrate how our scheme works to reproduce some standard (commutative) examples of T-dual spaces.

3.1 Twisted tori and their T-duals

We are interested in formulating a notion of T-duality for “torus bundles without H-flux”, which for our purposes can be characterised by the following general class of spaces.

Definition 3.1 (twisted tori). Let G be a locally compact group which admits a cocompact discrete subgroup ⇤G, i.e., a lattice in G, which we let act on G by left multiplication. The

quotient space T⇤G := ⇤G\G is a twisted torus.

By [34, Lemma 6.2], only unimodular groups can contain lattices, i.e., groups G whose mod-ular function G is identically equal to 1.

Example 3.2 (tori). Every lattice in the abelian Lie group G =Rd is isomorphic to ⇤G =Zd,

acting by translations. ThenT_Zd=Zd\Rd=:Td is the d-dimensional torus.

Example 3.3 (nilmanifolds). Generalizing Example 3.2, let G be a connected and simply-connected nilpotent Lie group. Then a theorem of Malcev [40, Theorem 2.12] establishes the existence of a lattice ⇤G in G if and only if G can be defined over the rationals, i.e., there exists

a basis for its Lie algebra which has rational structure constants, and in this caseT⇤G = ⇤G\G is a nilmanifold.

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Example 3.4 (orbifolds). Let G be a compact Lie group. Then the lattices in G are precisely the finite subgroups of G, andT = \G is a smooth orbifold. For instance, for G = SU(2) the twisted tori are precisely the three-dimensional ADE orbifoldsT = \ S3 of the three-sphere for a finite subgroup _{⇢ SU(2); for} =_Zn a cyclic subgroup of order n 2, this recovers the

familiar lens spacesT_Zn =Zn\ S

3 _=:_{L(n, 1).}

Following [32], we come now to a central concept of this paper.

Definition 3.5 (topological T-duality). LetT⇤G be a twisted torus which admits a non-trivial right action of the abelian Lie groupRnfor some n 1. The crossed product

C(_T⇤G)ortR n

is a C⇤-algebraic T-dual of the twisted torus.

If the spectrum of the crossed product algebra C(T⇤G)ort Rn is a Hausdor↵ topological space X (for instance if it is Morita equivalent to a commutative C⇤-algebra C(X)), then we say that X is T-dual to the twisted torusT⇤G and call X a ‘classical T-dual’; otherwise we say that the T-dual ofT⇤G is a noncommutative space.

For Definition3.5 to be a ‘good’ notion of T-duality, we should first explain (a) in what precise sense T⇤G and C(T⇤G)ortR

n _{are ‘equivalent’, and}

(b) how T-duality applied twice returns the original twisted torus.

The answers to both of these points turns out to be provided by working in a suitable category tailored to our treatment of topological T-duality.

3.2 T-duality in the category KK

The terminology ‘topological T-duality’ refers to a coarse equivalence at the level of topology; for C⇤-algebras the topology is measured by K-theory. A more powerful refinement is provided by Kasparov’s bivariant K-theory which constructs groups KK(_{A, B) for any pair of separable} C⇤-algebras_{A and B; when A = C, the group KK(C, B) ' K(B) is the K-theory group of B. The} cycles in Kasparov’s groups KK(A, B), called Kasparaov A–B bimodules, are triples (H, , T ) where_{H is a right Hilbert B-module, is a ⇤-representation of A on H, and T 2 End}_B(_{H) is a} B-linear operator on_{H, subject to certain compactness conditions; we do not provide further details} of the definition here and instead refer to [6] for a concise review of KK-theory in the context that we shall use it in this paper. Kasparov bimodules may be thought of as generalizations of morphisms between C⇤-algebras, in the sense that any algebra homomorphism :_{A ! B} determines a class [ ]2 KK(A, B), represented by the A–B-bimodule (B, , 0).

A key feature of Kasparov’s KK-theory is the composition product ⌦B: KK(A, B) ⇥ KK(B, C) ! KK(A, C),

which is bilinear and associative. This product is compatible with the composition of morphisms :_{A ! B and : B ! C of C}⇤-algebras: [ ]_⌦_B[ ] = [ ]. It also makes KK(_{A, A) into} a ring with unit 1_A = [id_A]. We say that an element ↵2 KK(A, B) is invertible if there exists an element 2 KK(B, A) such that ↵ ⌦B = 1A and ⌦A↵ = 1B.

An important special instance of Kasparov bimodules comes from Morita equivalence: Any Morita equivalence _{A–B bimodule M is also a Kasparov bimodule (M, , 0), with : A !} End(M) the left action of A, which defines an invertible class [M] 2 KK(A, B) with inverse [_{M ] 2 KK(B, A) given by the conjugate B–A bimodule M. Generally, if there exists an} invertible element ↵_{2 KK(A, B), then the algebras A and B are said to be KK-equivalent, and}

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we write _{A ⇠}KK B. Thus Morita equivalence implies KK-equivalence, but the converse is not generally true. KK-equivalent algebras have isomorphic K-theory groups, but not necessarily homeomorphic spectra.

This refinement naturally suggests an approach to T-duality where the category of separable C⇤-algebras with _{⇤-homomorphisms is replaced with an additive category KK , whose objects} are again separable C⇤-algebras but whose morphisms between any two objects_{A and B are given} by the classes in KK(A, B) (see, e.g., [7]). The composition product defines the composition law, and isomorphic algebras in this category are precisely the KK-equivalent algebras; in particular, Morita equivalent algebras are isomorphic as objects in KK . Our formulation and computations of topological T-duality will always take place in this category, and in this setting we can easily provide answers to points (a) and (b) below Definition3.5through

Theorem 3.6. If _T⇤G is a twisted torus with a non-trivial right action of R

n_{, then there are}

isomorphisms in the category KK given by the equivalences (a) C(T⇤G) ⇠KK C(T⇤G)ortR

n _{(up to a shift of degree n mod 2), and}

(b) C(T⇤G)ortR

n _o

b

rtRn ⇠M C(T⇤G).

Proof . The KK-equivalence (a) follows from the Connes–Thom isomorphism, formulated in the language of KK-theory [17]. The Morita equivalence (b) follows from Takai duality

(Theo-rem2.9). ⌅

Another virtue of the categorical setting is that it enables a general algebraic reformulation of the correspondence space construction, which for topological spaces ‘geometrizes’ the action of topological T-duality. In [7, Proposition 5.3] it is proven that, if A and B are separable C⇤

-algebras, then any class in KK(A, B) can be represented by a ‘noncommutative correspondence’. For this, we first recall, following [6,7], that KK-theory provides a definition of Gysin or “wrong way” homomorphisms on K-theory for C⇤-algebras. If :A ! B is a morphism of separable C⇤

-algebras, a K-orientation is a functorial assignment of a corresponding element !2 KK(B, A). If a K-orientation exists, we say that is K-oriented and call ! the associated Gysin element. The Gysin homomorphism on K-theory is now defined by !:= ( )⌦B ! : K(B) ! K(A). We

then slightly adapt the definition from [7] to the present context of Theorem3.6.

Definition 3.7 (noncommutative correspondences). LetT⇤G be a twisted torus which admits a non-trivial right action ofRn, and let

C C(_T⇤G) [ ] ;; C(_T⇤G)ortRn [ ] ff (3.1)

be a diagram in KK whose arrows are induced by homomorphisms : C(T⇤G) ! C and : C(T⇤G)ortRn ! C of separable C⇤-algebras. Assume that is K-oriented, and let ! 2 KK C, C(T⇤G)ortR

n _{be its corresponding Gysin element. The separable C}⇤_-algebra_{C is a}

non-commutative correspondence if the associated element [ ]⌦C !2 KK C(T⇤G), C(T⇤G)ortR

n

is a KK-equivalence between the twisted torus and its C⇤-algebraic T-dual.

Analogously to [4], we obtain a noncommutative correspondence by restricting theRn_-action

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Proposition 3.8. The crossed product C = C(T⇤G)ort|_Zn Z

n

is a noncommutative correspondence in the sense of Definition3.7.

Proof . We need to construct a diagram (3.1) in KK for the crossed product. For this, note that for any dynamical system of the form (_{A, ⇤, ↵) where ⇤ is a discrete group, there is a natural} injection j of the algebraA into the crossed product A o↵⇤: given a2 A, define the sequence

j(a)_{2 C}c(⇤,A) by j(a) = a ,e for 2 ⇤. It is easy to check, using the explicit formula for the

convolution product, that the map a_{7! j(a) is an algebra monomorphism: j(a) ? j(b) = j(a b)} for a, b2 A. In particular, there is a C⇤_{-algebra injection}

j : C(_T⇤G) ! C(T⇤G)ort|Zn Zn. (3.2)

Next we apply [20, Corollary 2.8] with Rn acting on Tn=Rn/Zn by (right) translation and the diagonal action ofRn onTn_{⇥ T}⇤G to obtain an isomorphism

C Tn⇥ T⇤G ortR n_{' C(T} ⇤G)ort|_Zn Z n _{⌦ K L}2 _Tn _. The projection_Tn_{⇥ T} ⇤G ! T⇤G induces an injection Cc(R n_{⇥ T} ⇤G) ,! Cc(R n_{⇥ T}n_{⇥ T} ⇤G) which preserves the convolution product, and we obtain a C⇤-algebra monomorphism

0_{: C(}_T

⇤G)ortR

n _{! C(T}

⇤G)ort|Zn Zn ⌦ K L2(Tn) ,

which is easily checked to be K-oriented since it is induced by a projection.

This gives algebra morphisms 0 := ◆ j : C(T⇤G)! C ⌦ K and 0: C(T⇤G)ortR

n_{! C ⌦ K,}

where _{C = C(T}⇤G)ort|Zn Zn, K denotes the C⇤-algebra of compact operators on a separable Hilbert space, and ◆ :C ! C ⌦K is the usual stabilization map. Taking the composition products of [ 0] and [ 0] with the Morita equivalence_{C⌦K ⇠}MC then yields the required maps in (3.1). ⌅ When the spectrum of the C⇤-algebraic T-dual is a Hausdor↵ space, we identify the corre-spondence space with the spectrum of_{C = C(T}⇤G)ort|_Zn Zn; otherwiseC is a noncommutative space.

3.3 Computational tools

Let us now explain how to compute these C⇤-algebraic T-duals in some special instances that will appear throughout the remainder of this paper. For certain actions ofR, we may compute the C⇤-algebraic T-dual via

Proposition 3.9. LetT⇤G be a twisted torus equipped with an action ofR for which every point has isotropy subgroup Z. Let T = T⇤G/R, and denote the corresponding principal circle bundle by p :T⇤G ! T . Then the C⇤-algebraic T-dual C(T⇤G)ortR ' CT(T ⇥ T, ) is a continuous-trace algebra with spectrum T ⇥ T and Dixmier–Douady class = ⇣ ^ c1(p) 2 H3(T ⇥ T, Z),

where c1(p)2 H2(T,Z) is the Chern class of the circle bundle and ⇣ is the standard generator

of H1(T, Z) ' Z.

Proof . This is just a straightforward adaptation of the statement of [39, Proposition 4.5]. ⌅ In these instances, the T-dual ofT⇤G is the Hausdor↵ space X =T ⇥ (T⇤G/R) with a three-form ‘H-flux’ whose cohomology class is represented by [H] = ⇣ ^ c1(p).

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More generally, suppose that the _Rn_{-action on} _T

⇤G is induced by a free and proper right action of Rn on the covering group G which commutes with the left action of the lattice ⇤G

on G. We can then apply Green’s theorem (Theorem2.13) to get the Morita equivalence C(T⇤G)ortR

n _⇠

M C0 G/Rn olt⇤G. (3.3)

In this special case, we obtain an easy proof of Proposition 3.8: The inclusion Zn ,_{! R}n of groups induces a monomorphism

C0 G/Rn olt⇤G ! C0 G/Zn olt⇤G ⇠M C(T⇤G)ort|_Zn Z n_,

where in the last step we replaced Rn _{by its subgroup} _Zn _{in (}_3.3_{). This gives}

monomor-phisms (3.2) and C(T⇤G)ortR

n_{! C(T}

⇤G)ort|ZnZn in the category KK .

3.4 Topological T-duality for the torus

Let us now describe how our considerations reproduce the standard T-duality for tori. The simplest example of the T-duality scheme (3.3) is the case where G = R, ⇤G = Z ⇢ R, and

G/R = {0} with the obviously trivial ⇤G-action. ThenTZ=Z\R = T is a circle, and (3.3) with

n = 1 reads

C(T) ortR ⇠M C o Z = C⇤(Z) ' C eT ,

where in the last passage we used the Fourier transform isomorphism _{F : C}⇤(Z) ! C eT ; ex-plicitly, if a =_{an}n2Z 2 C⇤(Z), then F(a)( ) =Pn2Zane2⇡in so that F(a) is a function on

the dual circle eT = R⇤/Z⇤.

The generalization to T-duality along a single direction i of a d-dimensional torus is straight-forward. Let ⇤ ' Zd _{be the lattice in} _Rd _{given by ⇤ =} Pd

i=1ai~ei| a1, . . . , ad 2 Z , where

~e1, . . . , ~ed is the standard basis of Rd; this is the direct sum ⇤ = Ldi=1Z~ei. Let Ri be the

subgroup of Rd linearly spanned by ~ei and let Zi ⇢ Ri be the corresponding lattice; we write

Ti = Ri/Zi and decompose the d-torus Td = Rd/⇤ as Td = Td 1_ˆı ⇥ Ti, where T_ˆıd 1 is the

(d 1)-dimensional torus defined by omitting the i-th factors ofRdand ⇤. ThenRi acts trivially

onT_ˆıd 1 and we have

C Td ortRi = C T_ˆıd 1⇥ Ti ortRi' C T_ˆıd 1 ⌦ C(Ti) oid⌦rtRi

' C Tˆıd 1 ⌦ C(Ti)ortRi ⇠M C T_ˆıd 1 ⌦ C eTi

' C Tˆıd 1⇥ eTi = C T˜ıd , (3.4)

whereT_˜ıd:=Td 1_ˆı _{⇥ e}Ti. This is the expected action of the i-th factorized T-duality, and in this

way we have thus reproduced the standard rules for T-duality of tori. In fact, in this case we can use Proposition3.9to strengthen the statement of topological T-duality: Every point ofTd has isotropy groupZ under the action of Ri, and the corresponding circle bundle p :Td! Td 1_ˆı

is trivial, so the C⇤-algebraic T-dual of _Td_{is a continuous-trace algebra with spectrum} _Td ˜ı and

trivial Dixmier–Douady class. Hence the Morita equivalence in (3.4) can be replaced by a stable isomorphism.

By iterating these T-duality transformations one can perform T-dualities along multiple directions of a d-dimensional torus. In particular, iterating the procedure d times and using Theorem2.6we end up with the full T-duality

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where e_Td₌ _Rd ⇤_/⇤⇤ _{is the dual torus with ⇤}⇤ _{the dual lattice in the dual vector space} _Rd ⇤_.

This can also be obtained directly by setting G = Rd, ⇤G = ⇤⇢ Rd and G/Rd = {0} in (3.3)

with n = d, and by using the Fourier transforms in all directions ~e1, . . . , ~ed.

Finally, let us consider the correspondence space construction. For the i-th factorized T-duality, this is obtained by restricting the action ofRi to the lattice Zi⇢ Ri. The action of the

groupZi on the algebra of functions C Td is trivial and so we get isomorphisms

C Td ortZi ' C Td ⌦ C⇤ Zi ' C Td ⌦ C eTi ' C Td⇥ eTi .

This results in the noncommutative correspondence induced by the diagram C Td_{⇥ e}Ti C Td pr⇤ 99 C T_˜ıd ⇡_i⇤ ee C T_ˆıd 1 j 99 j ee

where pr : Td⇥ eTi ! Td is the projection to the first factor and ⇡i:Td⇥ eTi ! T_˜ıd omits the

i-th factor ofTd. The algebra inclusions j of C T_ˆıd 1 are induced by the trivial circle bundle projectionsTd_{! T}d 1

ˆı and T˜ıd! Tˆıd 1.

By either iterating this construction using Theorem 2.6 or by direct calculation, the corre-spondence space for a full T-duality is obtained by restricting the action ofRd from (3.5) to the lattice ⇤_{⇢ R}d_{, and we analogously find}

C _Td _ort⇤' C Td⇥ eTd .

Thus the crossed product with the lattice of periods ⇤ defining the d-torus Td recovers the doubled torusTd_{⇥ e}_Td _{which is the correspondence space for the smooth Fourier–Mukai}

trans-form, wherein a full or factorized T-duality has a geometric interpretation as an element of its automorphism group GL(2d,Z).

3.5 Topological T-duality for orbifolds

Let G be a compact connected semisimple Lie group of rank r, and let _{⇢ G be a finite subgroup.} The maximal torus T = U(1)r' Rr_/_Zr _{of G carries a natural action of}_R

i by translation along

the i-th direction for i = 1, . . . , r, and we can apply a fibrewise T-duality to the principal torus bundle G_{! G/T. Under this R-action every point of G has isotropy subgroup Z, and the action} descends to the smooth orbifoldT = \G. Then the quotient map pi:T ! T /Ri is the circle

fibrationT ! T /U(1)i, where T = U(1)_ˆır 1⇥ U(1)i, and by Proposition 3.9the C⇤-algebraic

T-dual

C( _{\G) o}rtRi' CT \G/U(1)i⇥ eTi, i (3.6)

of the orbifold T is a continuous-trace algebra with spectrum T /U(1)i⇥ eTi and Dixmier–

Douady class i = c1(pi) ^ ⇣i.

In the rank one case, this T-duality is well-known (see, e.g., [3]): Then G = SU(2) which we regard as the three-sphere S3, and for = Zn ⇢ T = U(1) the twisted torus is the lens

space L(n, 1). The quotient map p: L(n, 1) ! S2 is a circle bundle whose Chern class c1(p) is

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C⇤-algebraic T-dual of _{L(n, 1) is a continuous-trace algebra whose spectrum is the trivial circle} bundle L(0, 1) = S2⇥eT and whose Dixmier–Douady class is n times the standard generator of H3 S2_⇥eT, Z ' Z.

Generally, the correspondence space construction is obtained by noting that, since the isotro-py subgroup for any point of theRi-action isZi ⇢ Ri, the group Zi acts trivially on the algebra

C(T ) and there are isomorphisms

C(T ) ortZi ' C(T ) ⌦ C⇤(Zi)' C(T ) ⌦ C eTi ' C T ⇥ eTi .

Let pr :T ⇥ eTi ! T be the projection to the first factor. Since H2(G,Z) = 0, K¨unneth’s

theorem implies pi⇥ ide_Ti

⇤ _c

1(pi) ^ ⇣i = 02 H3 T ⇥ eTi,Z ,

and hence the algebra CT T ⇥ eTi, (pi⇥ ide_Ti) ⇤

i is isomorphic to C T ⇥ eTi ⌦ K. Then there

is the noncommutative correspondence C T ⇥ eTi C(T ) [pr⇤_] 88 CT _{T /U(1)}i⇥ eTi, i [(pi⇥ide_Ti)⇤] ii C T /U(1)i [j] 55 [p⇤_] ff

as a diagram in the category KK .

4 Topological T-duality for almost abelian solvmanifolds

A large class of twisted tori of interest as string compactifications come in the form of fibrations over tori. These are the solvmanifolds which are based on solvable groups G and generalize the nilmanifolds discussed in Example3.3. The fibrations underlying these twisted tori are called Mostow bundles [35], and we are particularly interested in the cases where the Mostow bundle is a torus bundle. A good source for the material used in this section is [2] (see also [10,50]).

4.1 Mostow bundles

Let G be a connected and simply-connected solvable Lie group. Recall that its nilradical N is the maximal connected nilpotent normal subgroup. It has dimension dim N 1₂dim G.

We first consider the case dim N = dim G. Then N = G and the group G is nilpotent. In this case, under the conditions discussed in Example3.3, there exists a lattice ⇤G ⇢ G and the

twisted torus T⇤G is a nilmanifold. If G is abelian thenT⇤G is a torus. If G is non-abelian then there is a group extension

1 _{! [G, G] ! G} ⇡_{! G}ab ! 1

of its commutator subgroup [G, G], and both ⇤G\ [G, G] and ⇡(⇤G) are lattices in the nilpotent

Lie group [G, G] and the abelianization Gab := [G, G]\G of G, respectively [11]. This exhibits the

twisted torusT⇤G = ⇤G\G as a fibration over the torus ⇡(⇤G)\Gab with nilmanifold fibres, ⇤G\ [G, G] \[G, G] ! T⇤G ! ⇡(⇤G)\Gab.

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Suppose now that the group G is not nilpotent. Then N_{\G is a non-trivial abelian Lie group.} If G admits a lattice ⇤G, then ⇤N:= ⇤G\N is a lattice in N and ⇤GN = N⇤Gis a closed subgroup

of G, so ⇤GN\G is a torus. The twisted torus T⇤G = ⇤G\G is then a fibration over this torus with fibre the nilmanifold ⇤N\N = ⇤G\⇤GN. This bundle is called the Mostow bundle [35]. We

summarise these statements as

Theorem 4.1 (Mostow bundles). Let ⇤G be a lattice in a connected and simply-connected

solvable Lie group G andT⇤G = ⇤G\G the associated solvmanifold. Let N be the nilradical of G. Then ⇤GN is a closed subgroup of G, ⇤N := ⇤G\ N is a lattice in N, and ⇤GN\G is a torus. It

follows that the twisted torusT⇤G is a fibration over this torus with nilmanifold fibre: ⇤N\N = ⇤G\⇤GN ! T⇤G ! ⇤GN\G.

Remark 4.2. The structure group of the Mostow bundle is ⇤G0\⇤GN, where ⇤G0 is the largest subgroup of ⇤G which is normal in ⇤GN (cf. [2]). In particular, if ⇤G = ⇤G0 then the Mostow bundle is a principal ⇤G\⇤GN-bundle. In this case there is a well-defined left ⇤GN-action on

T⇤G = ⇤G\G and each point has isotropy subgroup ⇤G, so that the induced ⇤G\⇤GN-action is principal.

If the solvable Lie group G admits an abelian normal subgroup V, then ⇤GV = V⇤G is

a subgroup of G; if ⇤G is normal in ⇤GV, then the Mostow bundle construction can be refined via

an intermediate step involving a principal torus bundle over a second solvmanifold. Adapting [2, Theorem 3.6] we have

Proposition 4.3. Let G be a connected and simply-connected solvable Lie group and ⇤G a lattice

in G. Let V be a closed normal abelian Lie subgroup of G such that ⇤G is normal in V ⇤G. If

VZ:= ⇤G\ V is a lattice in V, then ⇤G\⇤GV is a torus and the solvmanifold T⇤G = ⇤G\G is the total space of the principal torus bundle

⇤G\⇤GV ! T⇤G ! ⇤GV\ G,

with base the solvmanifold T⇤_GV = ⇤GV\GV := VZ\⇤_G ✏ V\G = ⇤_GV\ G. There is moreover a double fibration Tn _//_T ⇤G ✏✏ ⇤_NV\NV //T_⇤ GV ✏✏ Tm (4.1)

where n = dim V, m = dim NV_\GV _{, N}V_{is the nilradical of G}V _{and ⇤}

NV = NV\⇤_GV the associated lattice.

Proof . Let p : G _{! V\G be the canonical projection. Since V is normal in G, and ⇤}G and

⇤G\ V are lattices in G and V, respectively, by [11, Lemma 5.1.4(a)] it follows that p(⇤G) is

a lattice in V\G. Hence p 1_(p(⇤

G)) = V⇤G = ⇤GV is closed in G, and ⇡ : G ! ⇤GV\G is a

bundle. By [49, Section 7.4] (adapted to the smooth case), since ⇤G is a closed normal subgroup

of ⇤GV, it follows that ⇤G\G ! ⇤GV\G is a principal ⇤G\⇤GV-bundle (or in other words,

Remark4.2holds as well under the present hypotheses). The fiber is a torus because V is abelian, ⇤G\⇤GV = VZ\V = Tn, with n = dim V. Moreover, because V is normal in G, there is a canonical

action of the group VZ_\⇤G on the connected and simply-connected solvable Lie group V\G

(given by VZ (Vg) = V( g) for 2 ⇤G and g2 G), so that ⇤GV\GV := VZ\⇤_G ✏

V\G is a solvmanifold. It is then easily proven that VZ_\⇤G ✏ V\G = ⇤GV\G. The double fibration (4.1)

Topological t-duality for twisted tori

Topological T-Duality for Twisted Tori

Contents

1

Introduction

2

Crossed products and duality

3

Topological T-duality and twisted tori

4

Topological T-duality for almost abelian solvmanifolds