Topological T-Duality for Twisted Tori
We apply the *-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 pr...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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Інститут математики НАН України
2021
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| Цитувати: | Topological T-Duality for Twisted Tori. Paolo Aschieri and Richard J. Szabo. SIGMA 17 (2021), 012, 51 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859994728149286912 |
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| author | Aschieri, Paolo Szabo, Richard J. |
| author_facet | Aschieri, Paolo Szabo, Richard J. |
| citation_txt | Topological T-Duality for Twisted Tori. Paolo Aschieri and Richard J. Szabo. SIGMA 17 (2021), 012, 51 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We apply the *-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 *-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 *-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 *-algebras rigorously describe the T-folds from non-geometric string theory.
|
| first_indexed | 2026-03-18T10:20:00Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 012, 51 pages
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, Università 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
Contents
1 Introduction 2
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Summary and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Crossed products and duality 5
2.1 Dynamical systems and their crossed products . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Semi-direct products and group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Pontryagin duality and Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Morita equivalence and Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
This paper is a contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at https://www.emis.de/journals/SIGMA/Landi.html
mailto:paolo.aschieri@uniupo.it
mailto:R.J.Szabo@hw.ac.uk
https://doi.org/10.3842/SIGMA.2021.012
https://www.emis.de/journals/SIGMA/Landi.html
2 P. Aschieri and R.J. Szabo
3 Topological T-duality and twisted tori 16
3.1 Twisted tori and their T-duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 T-duality in the category KK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Computational tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Topological T-duality for the torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Topological T-duality for orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Topological T-duality for almost abelian solvmanifolds 22
4.1 Mostow bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Almost abelian solvmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 C∗-algebra bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Rn-actions on Mostow bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Ry-actions: Circle bundles with H-flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Rz-actions: noncommutative torus bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Three-dimensional solvmanifolds and their T-duals 35
5.1 Mostow bundles and SL(2,R) conjugacy classes . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Parabolic torus bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3 Elliptic torus bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Hyperbolic torus bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References 49
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 different 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 Hausdorff topological spaces. In these instances the
Topological T-Duality for Twisted Tori 3
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 Hausdorff 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 connections 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 tori Tn over a circle T 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 TΛG
as the quotient
of a locally compact group G by a lattice ΛG in G; this definition encompasses the Tn-bundles
over T discussed above, along with many other known examples from string theory. We re-
gard TΛG
as a ‘torus bundle without H-flux’, which is captured simply by the C∗-algebra of
4 P. Aschieri and R.J. Szabo
functions C(TΛG
). 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 tradeoff 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 Section 3 we give our definition of twisted tori TΛG
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 Section 4 we 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(TΛG
) 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 of Rn, and in particular Rn-actions which act non-trivially on the base T would take the
algebra out of the category RKKT. 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 different 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
solvmanifold TΛG
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 Section 4. 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-torus T3
with H-flux, and the basic noncommutative principal T2-bundle over T 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 the Z2 and Z6 elliptic conjugacy classes,
as well as to the Poincaré solvmanifolds. In particular, our formalism extends the C∗-algebraic
Topological T-Duality for Twisted Tori 5
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 Hausdorff.
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 (γ, x) 7→ γ · 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, α) consisting of an algebra A and a locally compact group G acting
on A via a group homomorphism α : G → Aut(A), denoted γ 7→ (αγ : A → A) for γ ∈ G. In
topological T-duality one usually requires A to be a C∗-algebra, in which case (A,G, α) is called
a C∗-dynamical system. Two dynamical systems (A,G, α) and (B,G, β) are equivalent if there
is an algebra isomorphism ϕ : A → B which intertwines the G-actions: ϕ ◦ αγ = βγ ◦ ϕ for all
γ ∈ G.
If A is a commutative C∗-algebra, then we call (A,G, α) 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 α†γ
∣∣
X
for γ ∈ G under
the identification of points x ∈ 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, α), where αγ(f)(x) = f
(
γ−1 · x
)
for
γ ∈ G, f ∈ C0(X) and x ∈ 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, α) 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, α) in a C∗-algebra B
with multiplier algebra M(B) is a pair (Π,U) consisting of a homomorphism Π : A → M(B) and
a unitary representation U : G→ M(B), γ 7→ Uγ which satisfy the compatibility condition
Π
(
αγ(a)
)
= UγΠ(a)U−1
γ ,
for all γ ∈ G and a ∈ 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 Π : 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 on H; in this case we call (Π,U) a covariant representation of (A,G, α) 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 effective algebraic
“quotient” is through the crossed product algebra Aoα G for a dynamical system (A,G, α). For
a transformation group (X,G), this description is particularly powerful in the cases where the
6 P. Aschieri and R.J. Szabo
quotient X/G is not a Hausdorff 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
‖f‖univ := sup
(Π,U)
∥∥(Π oα U)(f)
∥∥
for compactly supported functions f ∈ Cc(G,A), where the supremum is taken over (possibly
degenerate) covariant representations (Π,U) of (A,G, α) with
(Π oα U)(f) :=
∫
G
Π
(
f(γ)
)
Uγ dµG(γ),
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 oα G is the completion (in
the universal norm) of the algebra Cc(G,A) equipped with the convolution product
(f ? f ′)(γ) :=
∫
G
f(γ′)αγ′
(
f ′
(
γ′−1γ
))
dµG(γ′), (2.1)
for all f, f ′ : 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 †(γ) := ∆G(γ)−1αγ
(
f
(
γ−1
)∗)
,
where ∆G : G→ R+ is the modular function of G defined through
∆G(γ′)
∫
G
f(γ) dµG(γ) =
∫
G
f(γγ′) dµG(γ)
for f ∈ Cc(G,A) and γ′ ∈ G. By the uniqueness of the left invariant Haar measure µG up to
a positive constant, ∆G(γ′) 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 = Cc(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 ? f ′)(γ, x) =
∫
G
f(γ′, x)f ′
(
γ′−1γ, γ′−1 · x
)
dµG(γ′),
while the ∗-algebra structure is given by
f †(γ, x) = ∆G(γ)−1f
(
γ−1, γ−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
αγ = idA : A → A for all γ ∈ G and (2.1) recovers the usual convolution product of functions on
the group G. As explained in Section 2.2 below, 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 algebra A, then Ao G ' A⊗ C∗(G).
The crossed product can be thought of as a universal object for covariant representations of
the dynamical system (A,G, α), in the following sense: Define the universal covariant represen-
tation (Π,U) of (A,G, α) in Aoα G by(
Π(a)f
)
(γ) = af(γ) and
(
Uγ′f
)
(γ) = αγ′
(
f
(
γ′−1γ
))
,
Topological T-Duality for Twisted Tori 7
for a ∈ A, f ∈ Cc(G,A) and γ, γ′ ∈ G. Then the universal property defining the crossed
product implies that any covariant representation (Π,U) of (A,G, α) in a C∗-algebra B fac-
tors through the universal covariant representation: There exists a unique homomorphism
ϕ : M(Aoα G)→ M(B) such that
Π = ϕ ◦Π and Uγ = ϕ(Uγ)
for all γ ∈ G.
If (Π,U) is a covariant representation of the dynamical system (A,G, α) on a Hilbert space H,
then
Φ(Π,U)(f) := (Π oα U)(f)
defines a representation Φ(Π,U) : Cc(G,A) → B(H) of the crossed product A oα G as bounded
operators on H. This is called the integrated form of the covariant representation (Π,U). In
particular, it maps the convolution product (2.1) onto the operator product in the algebra B(H),
Φ(Π,U)(f ? g) = Φ(Π,U)(f)Φ(Π,U)(g),
and it is covariant in the sense that
Φ(Π,U)
(
iG(γ)(f)
)
= UγΦ(Π,U)(f),
where
(
iG(γ)(f)
)
(γ′) := αγ
(
f
(
γ−1γ′
))
for each γ, γ′ ∈ G and f ∈ 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, τ θ
)
be the commutative C∗-dynamical system where τ θ is induced
through pullback by rotations of the circle T through a fixed angle θ ∈ R/Z:
τ θn(f)(z) = f
(
e 2πinθz
)
,
for n ∈ Z, f ∈ C(T) and z ∈ T. The resulting crossed product
Aθ := C(T) oτθ Z
is a called a rotation algebra, and for irrational values of θ it can be identified as a noncommu-
tative two-torus T2
θ in the following way.
By definition, the algebra Aθ is the universal norm completion of the convolution algebra
Cc(Z× T), whose elements f = {fn}n∈Z can be regarded as sequences (with only finitely many
nonvanishing terms) of functions fn : T→ C. The convolution product is given by
(f ?θ g)n(z) :=
∑
n′∈Z
fn′(z)gn−n′
(
e 2πin′θz
)
,
and the ∗-algebra structure is
f †n(z) := f−n
(
e 2πinθz
)
.
Via the Fourier transformation
f(z, w) :=
∑
n∈Z
fn(z)wn
8 P. Aschieri and R.J. Szabo
for w ∈ T, we may regard the convolution algebra Cc(Z × T) as a subspace of the space of
functions C
(
T2
)
equipped with the star-product
(f ?θ g)(z, w) =
∑
n∈Z
(f ?θ g)n(z)wn. (2.2)
After a further Fourier transformation
fn(z) =
∑
m∈Z
fm,nz
m
and some simple redefinitions of the Fourier series involved, the star-product (2.2) may be
written in the form
(f ?θ g)(z, w) =
∑
(m,n)∈Z2
( ∑
(m′,n′)∈Z2
fm′,n′gm−m′,n−n′ e
2πi(m−m′)n′θ
)
zmwn.
This recovers the usual commutative pointwise multiplication of functions in C
(
T2
)
for θ = 0.
For θ 6= 0 it realizes the irrational rotation algebra Aθ 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 of T2 whose star-product is a twisted convolution product on C
(
T2
)
.
In the language of covariant representations of the dynamical system (C(T),Z, τ θ), the crossed
product Aθ is the universal C∗-algebra generated by two unitaries U and V satisfying the rela-
tion [52, Proposition 2.56]
UV = e−2πiθV U. (2.3)
A concrete representation of Aθ on the Hilbert space H = L2(T) is given by defining
U(f)(z) = zf(z) and V (f)(z) = f
(
e 2πiθz
)
.
Example 2.2 (Noncommutative d-tori). The natural higher-dimensional generalization of
Example 2.1 involves a skew-symmetric real d×d matrix Θ = (θij), see [45]. Then the non-
commutative d-torus AΘ = TdΘ is the universal C∗-algebra generated by d unitaries U1, . . . , Ud
satisfying the relations
UiUj = e−2πiθijUjUi
for i, j = 1, . . . , d. By [37, Lemma 1.5], every noncommutative torus TdΘ can be obtained as
an iterated crossed product by Z in the following way. Let Θ|d−1 = (θij)1≤i,j≤d−1, and let
U1, . . . , Ud−1 be the standard generators of AΘ|d−1
= Td−1
Θ|d−1
. Define a group homomorphism
τ
~θ : Z→ Aut(AΘ|d−1
) by
τ
~θ
n(Ui) = e 2πinθidUi,
for n ∈ Z, where ~θ := (θid) ∈ Rd−1. Then there is an isomorphism of C∗-algebras
AΘ ' AΘ|d−1
o
τ~θ
Z. (2.4)
In the particular case where Θ|d−1 = 0d−1, we denote the corresponding noncommutative
d-torus by A~θ = Td~θ , 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 of Z:
A~θ ' C
(
Td−1
)
o
τ~θ
Z.
Topological T-Duality for Twisted Tori 9
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 = NH.
(2) Let N and H be two groups together with a left action ϕ : H → Aut(N) of H on N by
automorphisms, which we denote by ϕh(n) = hn for h ∈ H and n ∈ N; in particular
h(nn′) = hn hn′. We write HN to indicate that H acts on N from the left. The semi-direct
product of N and H is the group Noϕ H defined to be the set N× H with the product
(n, h) (n′, h′) =
(
n hn′, hh′
)
.
The inverse is then (n, h)−1 =
(
h−1
n−1, h−1
)
.
These two definitions are equivalent: Given subgroups N,H ⊂ G as in point (1), it follows that
G ' N oAd H where Ad is the adjoint action: Adh(n) = hnh−1. On the other hand, every
element of the group G = NoϕH defined in (2) can be written as (n, h) = (n, eH)(eN, h) and the
subgroups N× {eH} and {eN} × H intersect only in the identity of G. If the action of H on N is
trivial, i.e., ϕh = idN for all h ∈ H, then the semi-direct product reduces to the direct product
Noϕ 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 oϕ 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)oϕV H, where ϕV is
the action ϕ 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 to C[N];
equivalently it is the algebra of continuous functions on N with the convolution product. Given
a left H-action ϕ : H → Aut(N), there is an induced action ϕ∗ : 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) oϕ∗ H ' C∗(Noϕ H).
Proof. Note that C∗(N)oϕ∗ H = C[N]oϕ∗ H is the vector space of functions f : H→ C[N] with
convolution product(
f ?C[N]oϕ∗H f
′)(h) =
∑
h′∈H
f(h′) ?C[N] ϕ
∗
h′
(
f ′
(
h′−1h
))
,
and using the convolution product in C[N] this can be written as(
f ?C[N]oϕ∗H f
′)(n, h) =
∑
h′∈H
∑
n′∈N
f(n′, h′)f ′
(
h′−1(
n′−1n
)
, h′−1h
)
,
which is easily seen to coincide with the convolution product in C∗(Noϕ H) = C[Noϕ H]. �
10 P. Aschieri and R.J. Szabo
A more general result holds if N and H are locally compact groups with ϕ : H → Aut(N)
a continuous action of H on N via automorphisms (i.e., (h, n) 7→ ϕh(n) is a continuous map from
H × N to N). In this case the semi-direct product N oϕ 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 σ-compact, and N and H are closed subgroups of G.
The action β defining the C∗-dynamical system (C∗(N),H, β) and hence the crossed product
C∗(N) oβ H is the composition of the pullback ϕ∗ of the action ϕ : H→ Aut(N) with the action
σH : H→ R+ ⊂ Aut(C∗(N)) that enters the definition of the Haar measure on Noϕ 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 ) =
∫
N F
(
hn
)
dµN(n) for F ∈ C∗(N) is left invariant, i.e., Ih(λn′F ) = Ih(F ) where
(λn′F )(n) := F
(
n′−1n
)
for all n′ ∈ N (use invariance of the Haar measure under n → h−1
n′).
Uniqueness of the Haar measure up to a positive constant then implies there exists a function
σH : H→ R+ such that
σH(h)
∫
N
F
(
hn
)
dµN(n) =
∫
N
F (n) dµN(n). (2.5)
It is straighforward to see that σH is a group homomorphism and that it is continuous [52,
Section 2]. The Haar measure µNoϕH on Noϕ H is then given by∫
NoϕH
f(n, h) dµNoϕH(n, h) :=
∫
H
∫
N
f(n, h)σH(h)−1 dµN(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′)(n, h) =
(
h′n, h′h
)
, using (2.5) with F
(
h′n
)
:= f
(
h′n, h′h
)
and recalling that h is
fixed in (2.5).
Theorem 2.5. Let N and H be locally compact groups and ϕ : H→ Aut(N) a continuous action
of H on N. Define β : H → Aut(C∗(N)) by (βh′`)(n) = σH(h′)−1`
(
h′−1
n
)
for all h′ ∈ H and
` ∈ Cc(N). Then
C∗(N) oβ H ' C∗(Noϕ 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(N oϕ H) ↪→ Cc(N) oβ H, given by f(n, h) 7→ f(h) where f(h)(n) = f(n, h), the convolution
product in Cc(N oϕ H) is mapped to the convolution product in Cc(N) oβ H. Let f, f ′ ∈
Cc(Noϕ H), then(
f ?Cc(NoϕH) f
′)(n, h) =
∫
H
∫
N
f(n′, h′)f ′
(
(n′, h′)−1(n, h)
)
σH(h′)−1 dµN(n′) dµH(h′). (2.6)
On the other hand, for the images of f , f ′ in Cc(N) oβ H we have(
f ?Cc(N)oβH f
′)(h) =
∫
H
f(h′) ?C∗(N) βh′
(
f ′
(
h′−1h
))
dµH(h′).
Using the convolution product in C∗(N) this can be written as(
f ?Cc(N)oβH f
′)(h)(n) =
∫
H
∫
N
f(h′)(n′)βh′
(
f ′
(
h′−1h
))(
n′−1n
)
dµN(n′) dµH(h′),
which from the definition of the action β is easily seen to equal the image (f ?Cc(NoϕH) f
′)(h)(n)
in Cc(N) oβ H of the product (f ?Cc(NoϕH) f
′)(n, h) in Cc(Noϕ H) from (2.6).
More generally we have [52, Proposition 3.11]
Topological T-Duality for Twisted Tori 11
Theorem 2.6. Let (A,Noϕ H, α) be a C∗-dynamical system for the semi-direct product group
Noϕ H. Then (Aoα|N N,H, β) is a C∗-dynamical system, where
β : H −→ Aut(Aoα|N N), h 7−→ βh
is defined by
(
βh(f)
)
(n) = σH(h)−1αh
(
f
(h−1
n
))
for all f ∈ Cc(N,A) ⊂ Aoα|N N, with σH : H→
R+ defined by (2.5) and h−1
n = ϕh−1(n). Moreover, the canonical injection Cc(N oϕ H,A) ↪→
Cc
(
H, Cc(N,A)
)
extends to a C∗-algebra isomorphism
Aoα
(
Noϕ H
)
'
(
Aoα|N N
)
oβ H.
Theorem 2.5 is 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 Theorem 2.6.
Consider three groups M, N and H with group actions HN and NoHM; 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 actions HM, HN and NM satisfying the
compatibility conditions
h
(
nm
)
= (hn)
(
hm
)
for all m ∈ M, n ∈ N and h ∈ H. Then there exists a group action NoHM defined by (n,h)m =
n
(
hm
)
, and a group action H(Mo N) defined by h(m,n) =
(
hm, hn
)
, which together satisfy the
associativity property
Mo (No H) = (Mo N) o H.
2.3 Pontryagin duality and Fourier transform
If N is a locally compact abelian group we denote by N̂ its Pontryagin dual, i.e., the set of
characters χ : 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 N̂ = Rd and the
characters are given by χp(x) = e 2πi〈p,x〉 for x ∈ N and p ∈ N̂. The Pontryagin duality theorem
states that there is a canonical isomorphism
̂̂
N ' N, where n ∈ N is associated to the character
χ 7→ χ(n) on N̂.
The Fourier transform shows that the group C∗-algebra C∗(N) is isomorphic to C0
(
N̂
)
: Given
a Haar measure µN on N, the Fourier transform F(f) of f ∈ Cc(N) is defined by
F(f)(χ) :=
∫
N
f(n)χ(n) dµN(n)
for χ ∈ N̂. It sends the convolution product of functions in C∗(N) to the pointwise product of
functions in C(N̂):
F(f ? f ′) = F(f)F(f ′),
and extends to an isomorphism [52, Proposition 3.1]
F : C∗(N)
'−−→ C0
(
N̂
)
, (2.7)
where C0
(
N̂
)
is the algebra of functions on N̂ vanishing at infinity. For N separable, Hausdorff
and locally compact, C0
(
N̂
)
is indeed a C∗-algebra.
12 P. Aschieri and R.J. Szabo
Given a continuous left group action ϕ : H → Aut(N), which we also denote as before by
ϕh(n) = hn, consider the induced action β : H→ Aut
(
C∗(N)
)
as defined in Theorem 2.5. There
is also an induced left action ϕ̂ : H → Aut
(
N̂
)
defined by pullback:
(
ϕ̂hχ
)
(n) := χ
(
h−1
n
)
,
together with its pullback ϕ̂ ∗ : H → Aut
(
C0
(
N̂
))
defined by
(
ϕ̂ ∗h f̂
)
(χ) = f̂
(
ϕ̂h−1χ
)
for all
h ∈ H, f̂ ∈ C0
(
N̂
)
and χ ∈ N̂. The Fourier transform isomorphism (2.7) then extends to the
isomorphism
Proposition 2.8. If N is a locally compact abelian group and ϕ : H → Aut(N) is a continuous
action of a locally compact group H on N, then
C∗(N) oβ H ' C0
(
N̂
)
oϕ̂ ∗ H.
Proof. We show that the triples (C∗(N),H, β) and
(
C0
(
N̂
)
,H, ϕ̂ ∗
)
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 β and ϕ̂ ∗. For h ∈ H, f ∈ Cc(N) and χ ∈ N̂ we
compute
F
(
βh(f)
)
(χ) =
∫
N
βh(f)(n)χ(n) dµN(n) = σH(h)−1
∫
N
f
(
h−1
n
)
χ(n) dµN(n)
= σH(h)−1
∫
N
f
(
h−1
n
)
χ
(h(h−1
n
))
dµN(n) =
∫
N
f(n)χ
(
hn
)
dµN(n)
= F(f)
(
ϕ̂h−1χ
)
=
(
ϕ̂ ∗hF(f)
)
(χ),
where in the fourth equality we used (2.5) with F
(
h−1
n
)
= f
(
h−1
n
)
χ
(h(h−1
n
))
. �
Replacing N with N̂ in Proposition 2.8 we also obtain the isomorphism
C∗
(
N̂
)
o
β̂
H ' C0
(̂̂
N
)
ô̂ϕ ∗ H ' C0(N) oϕ∗ H, (2.8)
where β̂ : H → Aut
(
C∗(N̂)
)
is defined by β̂h(f̂ )(χ) = σ̂H(h)−1f̂
(
ϕ̂h−1χ
)
for h ∈ H, f̂ ∈ Cc(N̂)
and χ ∈ N̂, with σ̂H : H→ R+ defined as in (2.5) but using the dual group N̂ instead of N, and
in the final isomorphism we used Pontryagin duality
̂̂
N ' 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, α) is a C∗-dynamical system, then (Aoα G, Ĝ, α̂) is
a C∗-dynamical system, where
α̂ : Ĝ −→ Aut(Aoα G), χ 7−→ α̂χ
is defined by α̂χ(f)(γ) := χ(γ) f(γ) for all f ∈ Cc(G,A).
Theorem 2.9 (Takai duality). Let (A,G, α) be a C∗-dynamical system where G is a locally
compact abelian group. Then there is an isomorphism of C∗-algebras
(Aoα G) oα̂ Ĝ ' 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 · ξ) · b = a · (ξ · b)
Topological T-Duality for Twisted Tori 13
for all a ∈ A, b ∈ B and ξ ∈ 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〈 · , · 〉 and a B-valued inner product 〈 · , · 〉B satisfying the associativity condition
A〈ψ, φ〉 · ξ = ψ · 〈φ, ξ〉B,
for all ψ, φ, ξ ∈ M, under which M is complete in the norm closures, and such that the ideal
A〈M,M〉 is dense in A and 〈M,M〉B is dense in B. The bimodule M establishes a Morita
equivalence between the algebras A and B, and in this case we write A ∼M B.
Morita equivalent C∗-algebras have equivalent categories of nondegenerate ∗-representations:
If ΠB : B → B(HB) is a representation of B on a Hilbert space HB, then we can construct another
Hilbert space
HA :=M⊗B HB
which is the quotient of the tensor productM⊗HB by the relation (ξ · b)⊗ψ− ξ⊗ΠB(b)ψ = 0
identifying the B-actions for ξ ∈M, b ∈ B and ψ ∈ HB. The inner product on HA is given by〈
ξ ⊗B ψ
∣∣ξ′ ⊗B ψ′〉HA :=
〈
ψ
∣∣ΠB(〈ξ, ξ′〉B)ψ′〉HB ,
and a representation ΠA : A → B(HA) of the algebra A is defined by
ΠA(a)(ξ ⊗B ψ) = (a · ξ)⊗B ψ
for a ∈ A and ξ ⊗B ψ ∈ HA; this representation is unitary equivalent to the representation ΠB.
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 Aθ = T2
θ from Exam-
ple 2.1. Firstly, notice from (2.3) that changing the coset representative θ ∈ R/Z yields an
identical algebra: Aθ+m = Aθ for all m ∈ Z. Secondly, there is an obvious C∗-algebra isomor-
phism A−θ ' Aθ obtained by interchanging the two generators U and V . The converse is also
true [43, 44]: Aθ′ ' Aθ if and only if θ′ = θ mod 1. More generally, two irrational rotation
C∗-algebras Aθ and Aθ′ are Morita equivalent if and only if θ and θ′ lie in the same orbit under
the action of GL(2,Z) by fractional linear transformations
θ′ = M[θ] :=
aθ + b
cθ + d
for M =
(
a b
c d
)
∈ GL(2,Z).
The explicit Morita equivalence bimodules can be found in [43]. On the other hand, the rational
rotation algebras Aθ 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 Example 2.10 generalize
to the higher-dimensional noncommutative tori AΘ = TdΘ from Example 2.2 in the following
way [46]. First of all, the algebra AΘ is unchanged if the matrix Θ is written in another basis
of Zd: if B ∈ GL(d,Z) with transpose Bt, then there is a C∗-algebra isomorphism ABt ΘB ' AΘ.
More generally, consider the set of real skew-symmetric d×d matrices Θ whose orbits M [Θ] are
defined for all M ∈ SO(d, d;Z), where
M [Θ] = (AΘ +B) (C Θ +D)−1 for M =
(
A B
C D
)
∈ SO(d, d;Z),
14 P. Aschieri and R.J. Szabo
and A, B, C and D are d×d block matrices satisfying
AtC + CtA = 0 = BtD +DtB and AtD + CtB = 1d.
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 [Θ] ∼M AΘ.
However, for d > 2 the converse is not generally true: In fact, there are algebras AΘ and AΘ′
that are isomorphic (and so Morita equivalent) but for which the matrices Θ and Θ′ 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, α) and (B,G, β) be C∗-dynamical systems such that A and B are
Morita equivalent. Then the crossed product C∗-algebras AoαG and BoβG 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
Uγ(a · ξ) = αγ(a) · Uγ(ξ) and Uγ(ξ · b) = Uγ(ξ) · βγ(b),
and
A〈Uγ(ξ), Uγ(ξ′)〉 = αγ
(
A〈ξ, ξ′〉
)
and 〈Uγ(ξ), Uγ(ξ′)〉B = βγ
(
〈ξ, ξ′〉B
)
,
for all γ ∈ G, ξ, ξ′ ∈M, a ∈ A and b ∈ 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 ∈ C0(X), x ∈ X, h ∈ H and k ∈ 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 C0(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) ort H ∼M C0(X/H) olt K (2.9)
implemented by the Morita equivalence (or imprimitivity) bimodule M which is the completion
of Cc(X) with the actions
(a · ξ)(x) =
∫
K
a(k, x · H)ξ
(
k−1 · x
)
∆K(k)1/2 dµK(k),
(ξ · b)(x) =
∫
H
ξ
(
x · h−1
)
b
(
h,K · x · h−1
)
∆H(h)−1/2 dµH(h),
for all x ∈ X, a ∈ Cc(K×X/H), b ∈ Cc(H× K\X) and ξ ∈ Cc(X), and the inner products
A〈ξ, ξ′〉(k, x · H) = ∆K(k)−1/2
∫
H
ξ(x · h)ξ′
(
k−1 · x · h
)
dµH(h),
〈ξ, ξ′〉B(h,K · x) = ∆H(h)−1/2
∫
K
ξ
(
k−1 · x
)
ξ′
(
k−1 · x · h
)
dµK(k),
for all x ∈ X, h ∈ H, k ∈ K and ξ, ξ′ ∈ Cc(X).
Topological T-Duality for Twisted Tori 15
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) ort H ∼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) ort H ' C0(X/H)⊗K
(
L2(H)
)
, (2.10)
where K 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 ∈ Zd and x ∈ Rd, which realizes the d-dimensional
torus Td = Rd/Zd as a crossed product:
C
(
Td
)
∼M C0
(
Rd
)
ort Zd.
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}n∈Zd of functions fn : Rd → C with the
convolution product
(f ? g)n(x) =
∑
m∈Zd
fm(x)gn−m(x−m).
Consider the algebra
A :=
{
f ∈ C0
(
Rd,K
(
`2
(
Zd
))) ∣∣ f(x+m) = Umf(x)U−1
m
}
,
where Um is the unitary shift operator on `2
(
Zd
)
defined by (Uma)n = an−m for each m ∈ Zd
and a = {an}n∈Zd . Define a map Φ : Cc
(
Zd × Rd
)
→ Cc
(
Rd,K
(
`2
(
Zd
)))
by(
Φ(f)(x)
)
mn
= fm+n(x+ n).
It is easy to see that Φ(f)(x+m) = UmΦ(f)(x)U−1
m for all f ∈ Cc
(
Zd×Rd
)
, and if a = (amn) ∈ A
then defining fn := a0n gives Φ(f) = a. It is also easy to check that Φ(f ? g) = Φ(f)Φ(g), and
consequently Φ gives an algebra isomorphism
Φ : C0
(
Rd
)
ort Zd
'−−→ A.
The explicit Morita equivalence bimodule is now obtained from the completion of
M =
{
ξ ∈ Cc
(
Rd, `2
(
Zd
)) ∣∣ ξ(x+m) = Umξ(x)
}
.
The left action of the algebra A = Φ
(
Cc
(
Zd × Rd
))
is by left matrix multiplication on M:
A×M −→M,
(a, ξ) 7−→ a · ξ, (a · ξ)n =
∑
m∈Zd
anmξm,
while the right action of the algebra C
(
Td
)
is by right pointwise multiplication on M:
M× C
(
Td
)
−→M,
(ξ, b) 7−→ ξ · b, (ξ · b)n = ξnb.
16 P. Aschieri and R.J. Szabo
The left and right inner products are respectively given by
A〈ξ, η〉 = ξ ⊗ η∗,
〈ξ, η〉C(Td) =
∑
n∈Zd
ξnηn,
for ξ, η ∈ M. Together with the isomorphism Φ, this establishes a Morita equivalence between
the algebras C0
(
Rd
)
ort Zd 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 (ltγf)(x) = f
(
γ−1x
)
for all
f ∈ C0(G/H), x ∈ G/H and γ ∈ G), and the group C∗-algebra C∗(H) = Co H:
C0(G/H) olt G ∼M C∗(H).
This equivalence also follows from the C∗-algebra isomorphism [52, Theorem 4.29]
C0(G/H) olt G ' 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 ΛG, i.e., a lattice in G, which we let act on G by left multiplication. The
quotient space
TΛG
:= ΛG\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 ΛG = Zd,
acting by translations. Then TZd = 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 ΛG 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 case TΛG
= ΛG\G
is a nilmanifold.
Topological T-Duality for Twisted Tori 17
Example 3.4 (orbifolds). Let G be a compact Lie group. Then the lattices in G are precisely
the finite subgroups Γ of G, and TΓ = Γ\G is a smooth orbifold. For instance, for G = SU(2) the
twisted tori are precisely the three-dimensional ADE orbifolds TΓ = Γ\S3 of the three-sphere
for a finite subgroup Γ ⊂ SU(2); for Γ = Zn a cyclic subgroup of order n ≥ 2, this recovers the
familiar lens spaces TZn = Zn\S3 =: L(n, 1).
Following [32], we come now to a central concept of this paper.
Definition 3.5 (topological T-duality). Let TΛG
be a twisted torus which admits a non-trivial
right action of the abelian Lie group Rn for some n ≥ 1. The crossed product
C(TΛG
) ort Rn
is a C∗-algebraic T-dual of the twisted torus.
If the spectrum of the crossed product algebra C(TΛG
) ort Rn is a Hausdorff 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 torus TΛG
and call X a ‘classical T-dual’; otherwise we say
that the T-dual of TΛG
is a noncommutative space.
For Definition 3.5 to be a ‘good’ notion of T-duality, we should first explain
(a) in what precise sense TΛG
and C(TΛG
) ort Rn 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, φ, T )
where H is a right Hilbert B-module, φ is a ∗-representation of A on H, and T ∈ EndB(H) is a B-
linear operator onH, 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 φ : A → B
determines a class [φ] ∈ KK(A,B), represented by the A–B-bimodule (B, φ, 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
φ : A → B and ψ : B → C of C∗-algebras: [φ] ⊗B [ψ] = [ψ ◦ φ]. It also makes KK(A,A) into
a ring with unit 1A = [idA]. We say that an element α ∈ KK(A,B) is invertible if there exists
an element β ∈ KK(B,A) such that α⊗B β = 1A and β ⊗A α = 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, φ, 0), with φ : A →
End(M) the left action of A, which defines an invertible class [M] ∈ KK(A,B) with inverse
[M ] ∈ KK(B,A) given by the conjugate B–A bimodule M. Generally, if there exists an
invertible element α ∈ KK(A,B), then the algebras A and B are said to be KK-equivalent, and
18 P. Aschieri and R.J. Szabo
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 objectsA 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 Definition 3.5 through
Theorem 3.6. If TΛG
is a twisted torus with a non-trivial right action of Rn, then there are
isomorphisms in the category KK given by the equivalences
(a) C(TΛG
) ∼KK C(TΛG
) ort Rn (up to a shift of degree n mod 2), and
(b)
(
C(TΛG
) ort Rn
)
or̂t R
n ∼M C(TΛG
).
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-
rem 2.9). �
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 φ : A → B is a morphism of separable C∗-
algebras, a K-orientation is a functorial assignment of a corresponding element φ! ∈ KK(B,A).
If a K-orientation exists, we say that φ is K-oriented and call φ! the associated Gysin element.
The Gysin homomorphism on K-theory is now defined by φ! := (−)⊗B φ! : K(B) → K(A). We
then slightly adapt the definition from [7] to the present context of Theorem 3.6.
Definition 3.7 (noncommutative correspondences). Let TΛG
be a twisted torus which admits
a non-trivial right action of Rn, and let
C
C(TΛG
)
[φ]
;;
C(TΛG
) ort Rn
[ψ]
ff
(3.1)
be a diagram in KK whose arrows are induced by homomorphisms φ : C(TΛG
) → C and
ψ : C(TΛG
) ort Rn → C of separable C∗-algebras. Assume that ψ is K-oriented, and let ψ! ∈
KK
(
C, C(TΛG
)ortRn
)
be its corresponding Gysin element. The separable C∗-algebra C is a non-
commutative correspondence if the associated element
[φ]⊗C ψ! ∈ KK
(
C(TΛG
), C(TΛG
) ort Rn
)
is a KK-equivalence between the twisted torus and its C∗-algebraic T-dual.
Analogously to [4], we obtain a noncommutative correspondence by restricting the Rn-action
to the lattice Zn ⊂ Rn.
Topological T-Duality for Twisted Tori 19
Proposition 3.8. The crossed product
C = C(TΛG
) ort|Zn Zn
is a noncommutative correspondence in the sense of Definition 3.7.
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,Λ, α) where Λ is a discrete group, there is a natural
injection j of the algebra A into the crossed product Aoα Λ: given a ∈ A, define the sequence
j(a) ∈ Cc(Λ,A) by j(a)γ = aδγ,e for γ ∈ Λ. 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, b ∈ A. In particular, there is a C∗-algebra injection
j : C(TΛG
) −→ C(TΛG
) 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 of Rn on Tn × TΛG
to obtain an isomorphism
C
(
Tn × TΛG
)
ort Rn '
(
C(TΛG
) ort|Zn Zn
)
⊗K
(
L2
(
Tn
))
.
The projection Tn×TΛG
→ TΛG
induces an injection Cc(Rn×TΛG
) ↪→ Cc(Rn×Tn×TΛG
) which
preserves the convolution product, and we obtain a C∗-algebra monomorphism
ψ′ : C(TΛG
) ort Rn −→
(
C(TΛG
) 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 φ′ := ι ◦ j : C(TΛG
)→ C ⊗K and ψ′ : C(TΛG
) ort Rn → C ⊗K,
where C = C(TΛG
) ort|Zn Zn, K denotes the C∗-algebra of compact operators on a separable
Hilbert space, and ι : C → C⊗K is the usual stabilization map. Taking the composition products
of [φ′] and [ψ′] with the Morita equivalence C⊗K∼M C then yields the required maps in (3.1). �
When the spectrum of the C∗-algebraic T-dual is a Hausdorff space, we identify the corre-
spondence space with the spectrum of C = C(TΛG
) ort|Zn Zn; otherwise C 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 of R, we may compute
the C∗-algebraic T-dual via
Proposition 3.9. Let TΛG
be a twisted torus equipped with an action of R for which every point
has isotropy subgroup Z. Let T = TΛG
/R, and denote the corresponding principal circle bundle
by p : TΛG
→ T . Then the C∗-algebraic T-dual C(TΛG
) ort R ' CT(T × T, δ) is a continuous-
trace algebra with spectrum T × T and Dixmier–Douady class δ = ζ ^ c1(p) ∈ H3(T × T,Z),
where c1(p) ∈ H2(T,Z) is the Chern class of the circle bundle and ζ 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 of TΛG
is the Hausdorff space X = T× (TΛG
/R) with a three-
form ‘H-flux’ whose cohomology class is represented by [H] = ζ ^ c1(p).
20 P. Aschieri and R.J. Szabo
More generally, suppose that the Rn-action on TΛG
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 ΛG
on G. We can then apply Green’s theorem (Theorem 2.13) to get the Morita equivalence
C(TΛG
) ort Rn ∼M C0
(
G/Rn
)
olt ΛG. (3.3)
In this special case, we obtain an easy proof of Proposition 3.8: The inclusion Zn ↪→ Rn of
groups induces a monomorphism
C0
(
G/Rn
)
olt ΛG −→ C0
(
G/Zn
)
olt ΛG ∼M C(TΛG
) ort|Zn Zn,
where in the last step we replaced Rn by its subgroup Zn in (3.3). This gives monomor-
phisms (3.2) and C(TΛG
) ort Rn → C(TΛG
) ort|Zn Zn 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, ΛG = Z ⊂ R, and
G/R = {0} with the obviously trivial ΛG-action. Then TZ = Z\R = T is a circle, and (3.3) with
n = 1 reads
C(T) ort R ∼M Co Z = C∗(Z) ' C
(
T̃
)
,
where in the last passage we used the Fourier transform isomorphism F : C∗(Z) → C
(
T̃
)
; ex-
plicitly, if a = {an}n∈Z ∈ C∗(Z), then F(a)(χ) =
∑
n∈Z an e 2πinχ so that F(a) is a function on
the dual circle T̃ = R∗/Z∗.
The generalization to T-duality along a single direction i of a d-dimensional torus is straight-
forward. Let Λ ' Zd be the lattice in Rd given by Λ =
{∑d
i=1 ai~ei | a1, . . . , ad ∈ Z
}
, where
~e1, . . . , ~ed is the standard basis of Rd; this is the direct sum Λ =
⊕d
i=1 Z~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/Λ as Td = Td−1
ı̂ × Ti, where Td−1
ı̂ is the
(d−1)-dimensional torus defined by omitting the i-th factors of Rd and Λ. Then Ri acts trivially
on Td−1
ı̂ and we have
C
(
Td
)
ort Ri = C
(
Td−1
ı̂ × Ti
)
ort Ri '
(
C
(
Td−1
ı̂
)
⊗ C(Ti)
)
oid⊗rt Ri
' C
(
Td−1
ı̂
)
⊗
(
C(Ti) ort Ri
)
∼M C
(
Td−1
ı̂
)
⊗ C
(
T̃i
)
' C
(
Td−1
ı̂ × T̃i
)
= C
(
Tdı̃
)
, (3.4)
where Tdı̃ := Td−1
ı̂ × T̃i. 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 Proposition 3.9 to strengthen the statement of topological T-duality: Every point of Td
has isotropy group Z under the action of Ri, and the corresponding circle bundle p : Td → Td−1
ı̂
is trivial, so the C∗-algebraic T-dual of Td is a continuous-trace algebra with spectrum Tdı̃ 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
Theorem 2.6 we end up with the full T-duality
C
(
Td
)
ort Rd ∼M C
(
T̃d
)
, (3.5)
Topological T-Duality for Twisted Tori 21
where T̃d =
(
Rd
)∗
/Λ∗ is the dual torus with Λ∗ the dual lattice in the dual vector space
(
Rd
)∗
.
This can also be obtained directly by setting G = Rd, ΛG = Λ ⊂ 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 of Ri to the lattice Zi ⊂ Ri. The action of the
group Zi on the algebra of functions C
(
Td
)
is trivial and so we get isomorphisms
C
(
Td
)
ort Zi ' C
(
Td
)
⊗ C∗
(
Zi
)
' C
(
Td
)
⊗ C
(
T̃i
)
' C
(
Td × T̃i
)
.
This results in the noncommutative correspondence induced by the diagram
C
(
Td × T̃i
)
C
(
Td
) pr∗
99
C
(
Tdı̃
)
π∗i
ee
C
(
Td−1
ı̂
) j
99
j
ee
where pr : Td × T̃i → Td is the projection to the first factor and πi : Td × T̃i → Tdı̃ omits the
i-th factor of Td. The algebra inclusions j of C
(
Td−1
ı̂
)
are induced by the trivial circle bundle
projections Td → Td−1
ı̂ and Tdı̃ → Td−1
ı̂ .
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 of Rd from (3.5) to the
lattice Λ ⊂ Rd, and we analogously find
C
(
Td
)
ort Λ ' C
(
Td × T̃d
)
.
Thus the crossed product with the lattice of periods Λ defining the d-torus Td recovers the
doubled torus Td × T̃d 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 Γ ⊂ G be a finite subgroup.
The maximal torus T = U(1)r ' Rr/Zr of G carries a natural action of Ri 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 orbifold TΓ = Γ\G. Then the quotient map pi : TΓ → TΓ/Ri is the circle
fibration TΓ → TΓ/U(1)i, where T = U(1)r−1
ı̂ × U(1)i, and by Proposition 3.9 the C∗-algebraic
T-dual
C(Γ\G) ort Ri ' CT
(
Γ\G/U(1)i × T̃i, δi
)
(3.6)
of the orbifold TΓ is a continuous-trace algebra with spectrum TΓ/U(1)i × T̃i and Dixmier–
Douady class δi = c1(pi) ^ ζi.
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 Γ = 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
equal to n times the standard generator of H2
(
S2,Z
)
' Z, and applying (3.6) we find that the
22 P. Aschieri and R.J. Szabo
C∗-algebraic T-dual of L(n, 1) is a continuous-trace algebra whose spectrum is the trivial circle
bundle L(0, 1) = S2×T̃ and whose Dixmier–Douady class δ is n times the standard generator
of H3
(
S2×T̃,Z
)
' Z.
Generally, the correspondence space construction is obtained by noting that, since the isotro-
py subgroup for any point of the Ri-action is Zi ⊂ Ri, the group Zi acts trivially on the algebra
C(TΓ) and there are isomorphisms
C(TΓ) ort Zi ' C(TΓ)⊗ C∗(Zi) ' C(TΓ)⊗ C
(
T̃i
)
' C
(
TΓ × T̃i
)
.
Let pr : TΓ × T̃i → TΓ be the projection to the first factor. Since H2(G,Z) = 0, Künneth’s
theorem implies(
pi × idT̃i
)∗(
c1(pi) ^ ζi
)
= 0 ∈ H3
(
TΓ × T̃i,Z
)
,
and hence the algebra CT
(
TΓ × T̃i, (pi × idT̃i)
∗δi
)
is isomorphic to C
(
TΓ × T̃i
)
⊗K. Then there
is the noncommutative correspondence
C
(
TΓ × T̃i
)
C(TΓ)
[pr∗]
88
CT
(
TΓ/U(1)i × T̃i, δi
)
[(pi×idT̃i
)∗]
ii
C
(
TΓ/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 Example 3.3. 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 dimN ≥ 1
2 dimG.
We first consider the case dimN = dimG. Then N = G and the group G is nilpotent. In
this case, under the conditions discussed in Example 3.3, there exists a lattice ΛG ⊂ G and the
twisted torus TΛG
is a nilmanifold. If G is abelian then TΛG
is a torus. If G is non-abelian then
there is a group extension
1 −→ [G,G] −→ G
π−−→ Gab −→ 1
of its commutator subgroup [G,G], and both ΛG ∩ [G,G] and π(ΛG) are lattices in the nilpotent
Lie group [G,G] and the abelianization Gab := [G,G]\G of G, respectively [11]. This exhibits the
twisted torus TΛG
= ΛG\G as a fibration over the torus π(ΛG)\Gab with nilmanifold fibres,(
ΛG ∩ [G,G]
)
\[G,G] −→ TΛG
−→ π(ΛG)\Gab.
If [G,G] is an abelian Lie group then the twisted torus is a torus bundle over a torus.
Topological T-Duality for Twisted Tori 23
Suppose now that the group G is not nilpotent. Then N\G is a non-trivial abelian Lie group.
If G admits a lattice ΛG, then ΛN := ΛG∩N is a lattice in N and ΛGN = NΛG is a closed subgroup
of G, so ΛGN\G is a torus. The twisted torus TΛG
= ΛG\G is then a fibration over this torus
with fibre the nilmanifold ΛN\N = ΛG\ΛGN. This bundle is called the Mostow bundle [35]. We
summarise these statements as
Theorem 4.1 (Mostow bundles). Let ΛG be a lattice in a connected and simply-connected
solvable Lie group G and TΛG
= ΛG\G the associated solvmanifold. Let N be the nilradical of G.
Then ΛGN is a closed subgroup of G, ΛN := ΛG ∩ N is a lattice in N, and ΛGN\G is a torus. It
follows that the twisted torus TΛG
is a fibration over this torus with nilmanifold fibre:
ΛN\N = ΛG\ΛGN −→ TΛG
−→ ΛGN\G.
Remark 4.2. The structure group of the Mostow bundle is ΛG0\ΛGN, where ΛG0 is the largest
subgroup of ΛG which is normal in ΛGN (cf. [2]). In particular, if ΛG = ΛG0 then the Mostow
bundle is a principal ΛG\ΛGN-bundle. In this case there is a well-defined left ΛGN-action on
TΛG
= ΛG\G and each point has isotropy subgroup ΛG, so that the induced ΛG\ΛGN-action is
principal.
If the solvable Lie group G admits an abelian normal subgroup V, then ΛGV = VΛG is
a subgroup of G; if ΛG is normal in ΛGV, 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 ΛG a lattice
in G. Let V be a closed normal abelian Lie subgroup of G such that ΛG is normal in VΛG. If
VZ := ΛG ∩V is a lattice in V, then ΛG\ΛGV is a torus and the solvmanifold TΛG
= ΛG\G is the
total space of the principal torus bundle
ΛG\ΛGV −→ TΛG
−→ ΛGV \ G,
with base the solvmanifold TΛ
GV
= ΛGV\GV :=
(
VZ\ΛG
)∖(
V\G
)
= ΛGV \ G. There is moreover
a double fibration
Tn // TΛG
��
ΛNV\NV // TΛ
GV
��
Tm
(4.1)
where n = dimV, m = dim
(
NV\GV
)
, NV is the nilradical of GV and ΛNV = NV∩ΛGV the associated
lattice.
Proof. Let p: G → V\G be the canonical projection. Since V is normal in G, and ΛG and
ΛG ∩ V are lattices in G and V, respectively, by [11, Lemma 5.1.4(a)] it follows that p(ΛG) is
a lattice in V\G. Hence p−1(p(ΛG)) = VΛG = ΛGV is closed in G, and π : G → ΛGV\G is a
bundle. By [49, Section 7.4] (adapted to the smooth case), since ΛG is a closed normal subgroup
of ΛGV, it follows that ΛG\G → ΛGV\G is a principal ΛG\ΛGV-bundle (or in other words,
Remark 4.2 holds as well under the present hypotheses). The fiber is a torus because V is abelian,
ΛG\ΛGV = VZ\V = Tn, with n = dimV. Moreover, because V is normal in G, there is a canonical
action of the group VZ\ΛG on the connected and simply-connected solvable Lie group V\G
(given by
(
VZλ
)
(Vg) = V(λg) for λ ∈ ΛG and g ∈ G), so that ΛGV\GV :=
(
VZ\ΛG
)∖(
V\G
)
is a
solvmanifold. It is then easily proven that
(
VZ\ΛG
)∖(
V\G
)
= ΛGV\G. The double fibration (4.1)
follows immediately from Theorem 4.1 applied to TΛ
GV
= ΛGV\GV. �
24 P. Aschieri and R.J. Szabo
Remark 4.4. An equivalent form for the double fibration (4.1) is given by
Tn // TΛG
��
ΛNV\NV // TΛG
/V
��
Tm
which is obtained by observing that ΛGV\G = ΛG\G
/
V since V is normal in G. Notice also
that G/V = V\G and ΛG/V
Z = VZ\ΛG (because VZ = ΛG ∩ V is normal in ΛG), and moreover
ΛGV\GV =
(
VZ\ΛG
)∖
(V\G) =
(
ΛG/V
Z)∖(G/V).
4.2 Almost abelian solvmanifolds
In contrast to the case of nilpotent groups, there is no simple criterion for the existence of
a lattice in a general connected and simply-connected solvable Lie group, as is required to define
a corresponding twisted torus. To formulate such a criterion, we specialise to almost abelian
solvable groups: these are the solvable Lie groups G of dimension d whose nilradical N has
codimension one and is abelian:
N ' Rd−1.
Then G has the structure of a semi-direct product
G = Noϕ R
for a continuous one-parameter left group action ϕ : R→ Aut(N); concretely, ϕ is given by the
adjoint action of the one-dimensional subgroup H = R on N in the group G (cf. Section 2.2).
This exhibits G as a nontrivial group extension
1 −→ N −→ G −→ R −→ 1.
We can regard the one-parameter group action ϕ as a matrix ϕx ∈ GL(d−1,R) for each x ∈ R
acting on the vector space N, which we identify with Rd−1 via a choice of basis ~e1, . . . , ~ed−1. Since
ϕ0 = 1d−1 and ϕx is always non-singular, it follows from the continuity of ϕ and the determinant
that detϕx > 0 for all x ∈ R. Then G admits a lattice ΛG if and only if there exists x0 ∈ R×
such that ϕx0 is conjugate to an integer matrix M ∈ SL(d− 1,Z):
Σ−1ϕx0Σ = M (4.2)
for some Σ ∈ GL(d−1,R). In this case the twisted torus TΛG
= ΛG\G is called an almost abelian
solvmanifold . The condition (4.2) strongly restricts the homomorphisms ϕ : R → Aut(N); in
particular, it requires that the characteristic polynomial of ϕx0 has integer coefficients. In this
case the lattice (in the standard basis ~e1, . . . , ~ed−1) is given by
ΛG = Σ · Zd−1 oϕ|x0 Z x0Z,
which correspondingly sits as a nontrivial group extension
1 −→ Σ · Zd−1 −→ ΛG −→ x0Z −→ 1.
Then ΛN \ N ' Td−1, and the corresponding Mostow bundle realises the twisted torus TΛG
as
a torus bundle over a circle ΛGN\G ' T, whose monodromy is specified by the matrix M in the
Topological T-Duality for Twisted Tori 25
mapping class group SL(d − 1,Z) of orientation-preserving automorphisms up to homotopy of
the torus fibres Td−1.
For an almost abelian solvmanifold we can make the twisted torus construction more concrete
by choosing the global coordinates (z, x) ∈ Rd−1 × R on the group manifold (associated with
the basis ~e1, . . . , ~ed−1). The group multiplication of the semi-direct product G = Rd−1 oϕ R is
(z, x)(z′, x′) = (z + ϕx · z′, x+ x′), (4.3)
where we used ϕxϕx′ = ϕx+x′ , and the inverse of a group element is
(z, x)−1 = (−ϕ−x · z,−x),
where we used ϕ−1
x = ϕ−x. The twisted torus is defined as the quotient TΛG
= ΛG\G which is
generated by the equivalence relation (z, x) ∼ (Σ · γ, x0 α) (z, x) for all (γ, α) ∈ Zd−1 × Z. The
global structure of TΛG
is generated by the simultaneous local coordinate identifications under
the action of the elements (Σ · γ, x0α) of ΛG given by
(z, x) 7−→ (z + Σ · γ, x),
(z, x) 7−→ (ΣMα Σ−1 · z, x+ x0 α). (4.4)
These identifications explicitly exhibit the twisted torus as a torus bundle over a circle, with
local fiber coordinates z ∈ Td−1 and base coordinate x ∈ T, whose monodromy is specified by
the matrix M ∈ SL(d− 1,Z), and whose periods are given respectively by Σ ∈ GL(d− 1,R) and
x0 ∈ R×.
4.3 C∗-algebra bundles
Mostow bundles and their C∗-algebraic T-duals can be grouped together under the general
heading of ‘C∗-algebra bundles’, which encompasses the notions of C0(X)-algebras and C∗-
bundles, as we now explain; see Section 8.1 and Appendix C of [52] for further details. Let X be
a locally compact Hausdorff space. There are two equivalent notions for the C∗-algebra analogue
of a fibre bundle over X.
A C0(X)-algebra is a C∗-algebra A equipped with a nondegenerate injection ι of C0(X) into
the centre of its multiplier algebra, called the structure map. For f ∈ C0(X) and a ∈ A, we
abbreviate ι(f)a by f · a. This endows A with a C0(X)-bimodule structure.
Given a family B = (Bx)x∈X of C∗-algebras, a section of B is a map s : X → B such that
s(x) ∈ Bx for all x ∈ X; we denote the space of sections of B which vanish at infinity by Γ0(B).
The family B is then called a C∗-bundle over X with fibres Bx if the following conditions are
satisfied:
� Γ0(B) is a C∗-algebra under pointwise operations and the supremum norm;
� Bx = {s(x) | s ∈ Γ0(B)} for each x ∈ X;
� Γ0(B) is closed under multiplication by C0(X); and
� For each s ∈ Γ0(B), the function x 7→ ‖s(x)‖ is upper semi-continuous, i.e., the set
{x ∈ X | ‖s(x)‖ < ε} is open in X for all ε > 0.
In this paper we will only be concerned with C∗-bundles that have non-zero fibres.
If B is a C∗-bundle over X, then its section algebra Γ0(B) is a C0(X)-algebra: its structure
map ι from C0(X) is defined by ι(f)s = fs. Conversely, if A is a C0(X)-algebra, then the
fibre Ax of A over x ∈ X is Ax := A/Ix, where Ix = {f · a | f ∈ C0(X), f(x) = 0, a ∈ A} is
26 P. Aschieri and R.J. Szabo
identified as the ideal in A of sections vanishing at x. If a ∈ A, we write a(x) = a + Ix for its
image in Ax. The function x 7→ ‖a(x)‖ is upper semi-continuous and vanishes at infinity with
‖a‖ = sup
x∈X
∥∥a(x)
∥∥
for all a ∈ A. The elements a ∈ A can in this way be viewed as sections of a C∗-bundle (Ax)x∈X .
We will sometimes use the notation
∐
x∈X Ax for A when we wish to emphasise its structure as
a C∗-bundle over X with fibre C∗-algebras Ax. These definitions do not require local triviality
of the bundle nor the fibres of the bundle to be isomorphic to one another. C∗-algebra bundles
over X are objects of a category whose morphisms are fibrewise ∗-homomorphisms, i.e., C0(X)-
linear morphisms ψ : A → B: ψ(f · a) = f · ψ(a) for all f ∈ C0(X) and a ∈ A; then ψ induces
∗-homomorphisms ψx : Ax → Bx such that ψx
(
a(x)
)
= ψ(a)(x) for all a ∈ A.
Example 4.5 (trivial C∗-algebra bundles). If D is any C∗-algebra, then A = C0(X,D) '
C0(X)⊗D is naturally a C0(X)-algebra with structure map(
ι(f)a
)
(x) := f(x) a(x)
for f ∈ C0(X), a ∈ A and x ∈ X. In this case each fibre Ax is canonically identified with D and
elements of A are obviously identified with sections.
Example 4.6 (continuous maps). Let X and Y be locally compact spaces and σ : Y → X
a continuous surjective map. Then C0(Y ) is a C0(X)-algebra with structure map ι(f)g :=
(f ◦ σ) g, for f ∈ C0(X) and g ∈ C0(Y ), and fibers C0(Y )x ' C0
(
σ−1(x)
)
.
In this paper we are particularly interested in crossed products of C∗-algebra bundles. Let A
be a C0(X)-algebra, and denote by AutX(A) the group of fibrewise automorphisms of A. A fibre-
wise action of a locally compact group G on A is then a group homomorphism α : G→ AutX(A).
This implies that α induces an action αx on each fiber Ax for x ∈ X, and in this case we say
that the dynamical system (A,G, α) is C0(X)-linear.
Theorem 4.7. Let X be a locally compact Hausdorff space, and let (A,G, α) be a C0(X)-linear
C∗-dynamical system. Then the crossed product Aoα G is again a C0(X)-algebra with fibres
(Aoα G)x ' Ax oαx G,
where
αxγ
(
a(x)
)
= αγ(a)(x)
for each x ∈ X, γ ∈ G and a ∈ A.
Proof. The structure map of A oα G is given by precomposing the structure map of A with
the natural injection of the center of the multiplier algebra of A into the center of the multiplier
algebra of Aoα G; it satisfies (f · f)(γ) = f ·
(
f(γ)
)
for all f ∈ C0(X), f ∈ Cc(G,A) and γ ∈ G.
See [52, Theorem 8.4] for further details. �
Example 4.8 (transformation groups). Let (X,G) be a second countable transformation group
whose quotient X/G is a Hausdorff space. By Example 4.6, C0(X) is a C0(X/G)-algebra whose
fiber over G · x is isomorphic to C0(G/Gx), where Gx = {γ ∈ G | γ · x = x} is the stabilizer
subgroup at x ∈ X. Then the crossed product C0(X)oαG is the section algebra of a C∗-algebra
bundle over X/G whose fiber over G · x is isomorphic to C∗(Gx) ⊗ K
(
L2(G/Gx)
)
[51]. In the
special case where G = R and Gx = Z for all x ∈ X, this is contained in the statement of
Proposition 3.9.
Topological T-Duality for Twisted Tori 27
Example 4.9 (principal torus bundles). Let E → X be a principal Tr-bundle. By Exam-
ple 4.8, C0(E) is a C0(X)-algebra with fibers C0(E)x ' C(Tr), and by (2.10) there is a stable
isomorphism C0(E) ort Tr ' C0(X)⊗K
(
L2(Tr)
)
. More generally, a noncommutative principal
Tr-bundle on X is a C0(X)-linear C∗-dynamical system (A,Tr, α) with an isomorphism
Aoα Tr ' C0(X,K)
of C∗-algebra bundles over X. For further details and a classification of noncommutative prin-
cipal torus bundles, see [15, 21].
Example 4.10 (noncommutative correspondences). The noncommutative correspondence C =
C(TΛG
) ort Zn from Proposition 3.8 is a noncommutative principal Tn-bundle on X = TΛG
in the sense of Example 4.9: The C∗-algebra C is naturally equipped with the dual action of
Tn = Ẑn, and the Takai duality theorem implies that there is an isomorphism C or̂t T
n '
C(TΛG
)⊗K
(
`2(Zn)
)
of C∗-algebra bundles over the twisted torus TΛG
.
Remark 4.11. There is a natural notion of Morita equivalence of C∗-algebra bundles over X,
similar to the notion of equivariant Morita equivalence from Theorem 2.12, which uses the
C0(X)-bimodule structures: a C0(X)-linear Morita equivalence between two C0(X)-algebras is
a Morita equivalence which is compatible with the C∗-bundle structures over X. More gener-
ally, there is a category RKKX of C∗-algebra bundles over X whose morphisms are elements
of Kasparov’s groups RKK(X;A,B), see, e.g., [15]: the cycles are the usual cycles (H, φ, T )
for Kasparov’s bivariant K-theory KK(A,B) (cf. Section 3.2) with the additional requirement
that φ : A → EndB(H) is C0(X)-linear. There is an obvious faithful functor RKKX → KK
which forgets the C0(X)-algebra structures. Isomorphic C∗-bundles in the category RKKX are
precisely the RKK-equivalent C∗-bundles. If A and B are isomorphic in RKKX , i.e., there
exists an invertible class α ∈ RKK(X;A,B), then they are also isomorphic as C∗-algebras in
the category KK .
4.4 Rn-actions on Mostow bundles
We can now apply the results of Section 3 to the class of twisted tori given in Section 4.2.
The Mostow fibration of any almost abelian solvmanifold identifies TΛG
as a torus bundle over
a circle, hence the algebra of functions C(TΛG
) is a C(T)-algebra. In other words, C(TΛG
) is an
object of the category RKKT, and we are interested in the T-duality isomorphisms of C(TΛG
) in
this category. In particular, given a fibrewise right action of the abelian Lie group Rn on TΛG
, it
follows from Theorem 4.7 that the C∗-algebraic T-dual C(TΛG
)ortRn is also a C(T)-algebra, and
by [15, Theorem 3.5] the C∗-bundles C(TΛG
) and C(TΛG
) ort Rn are isomorphic as C∗-algebras
in the category RKKT.
In order to have sensible definitions of T-duality, we need to identify the homologically non-
trivial one-cycles of the twisted torus TΛG
, which are determined in [2, Proposition 4.7]. Write
M = (mij)
for the integer matrix elements mij ∈ Z of the monodromy matrix. Since G = Rd−1 oϕ R is
simply-connected, the fundamental group of the twisted torus is π1(TΛG
) ' ΛG whose abeliani-
sation ΛG/[ΛG,ΛG] gives the first homology group via the presentation
H1(TΛG
,Z) = Z⊕
〈
ê1, . . . , êd−1
∣∣∣ d−1∑
j=1
mjiêj = êi for i = 1, . . . , d− 1
〉
, (4.5)
28 P. Aschieri and R.J. Szabo
where the first factor of Z corresponds to the base circle of the torus fibration and the generators
ê1, . . . , êd−1 correspond to the torus fibres; they are given by
êi :=
d−1∑
k=1
Σki~ek, (4.6)
where ~e1, . . . , ~ed−1 is the standard basis of Zd−1 giving the group law (4.3).
We can then apply the structure theorem for finitely-generated Z-modules by appealing to
some classical matrix algebra. From the presentation (4.5), H1(TΛG
,Z) = Z⊕coker(ϕx0− idZd−1)
where (ϕx0−idZd−1) : Zd−1 → Zd−1 in the basis ê1, . . . , êd−1 is given by the integer relation matrix
A := M− 1d−1 . (4.7)
Let r be the rank of A. This matrix can be brought into its Smith normal form D by finding
invertible integer matrices L, R ∈ GL(d− 1,Z) such that
D = LAR
is diagonal with entries mi ∈ Z for i = 1, . . . , d− 1. The integers mi are the elementary divisors
of A. They have the properties that mi divides mi+1, for 0 < i < d− 1, and in particular mi = 0
for i > r; they can be computed explicitly (up to sign) as
mi =
di(A)
di−1(A)
,
where the i-th determinant divisor di(A) is the greatest common divisor of all i×i minors of
the relation matrix A, with d0(A) := 1. The matrices L, R ∈ GL(d − 1,Z) are found by reducing
the matrix A to its Smith normal form D through a sequence of elementary row and column
operations over Z, see, e.g., [23].
Given the Smith normal form, we set
ẽi :=
d−1∑
j=1
(
L−1
)
ji
êj (4.8)
and observe that the image of A, which is generated over Z by the vectors
d−1∑
j=1
Ajiêj =
d−1∑
k,l=1
(
R−1
)
ki
Dlkẽk,
is equivalently generated by the vectors mkẽk with k = 1, . . . , r. Hence
coker(ϕx0 − idZd−1) =
〈
ê1, . . . , êd−1
〉/〈 d−1∑
j=1
Aj1êj , . . . ,
d−1∑
j=1
Ajd−1êj
〉
=
〈
ẽ1, . . . , ẽd−1
〉/〈
m1ẽ1, . . . ,mrẽr
〉
and
H1(TΛG
,Z) ' Z⊕ Zd−1−r ⊕
r⊕
i=1
Zmi . (4.9)
We are exclusively interested in the natural Rn-actions on TΛG
which descend from actions
of abelian subgroups Rn ⊂ G, acting on G by right multiplication. They can be organised into
Topological T-Duality for Twisted Tori 29
three classes associated with the different types of summands in the Z-module presentation of
the homology group (4.9), and we only retain those which are fiberwise actions on the Mostow
bundle. The first summand Z in (4.9) corresponds to the subgroup
Rx =
{
(0, ξ) ∈ G
}
acting on G by right multiplication:
(z, x)(0, ξ) = (z, x+ ξ)
for all (z, x) ∈ G and ξ ∈ R. Clearly this does not descend to a fiberwise action on the twisted
torus TΛG
, and the crossed product C(TΛG
) ort Rx is no longer a C(T)-algebra. Thus an Rx-
action takes us out of the category RKKT, and we will henceforth discard Rn-actions where Rn
contains the subgroup Rx.
For the remaining types of summands in (4.9), we can give explicit descriptions of the C∗-
algebraic T-duals of an almost abelian solvmanifold. We consider both classes in turn. As the
only solvmanifolds in one and two dimensions are tori, which are already treated by our analysis
from Section 3.4, we assume d ≥ 3 for the remainder of this paper.
4.5 Ry-actions: Circle bundles with H-flux
Let us consider the second summand Zd−1−r in (4.9), which corresponds to the lattice ΛG ∩
ker
(
ϕx0 − idRd−1
)
in ker
(
ϕx0 − idRd−1
)
. In terms of the generators
e′i :=
d−1∑
j=1
Rjiêj
of ΛG ∩N, the sublattice ΛG ∩ ker
(
ϕx0 − idRd−1
)
is generated by the vectors e′r+1, e
′
r+2, . . . , e
′
d−1
in the kernel of the relation matrix (4.7). We begin with some elementary observations. Firstly,
the subgroups ker
(
ϕx0 − idRd−1
)
and ΛG commute in G: (−v, 0)(0, x0)(v, 0) = (0, x0) ∈ ΛG for
all v ∈ ker
(
ϕx0− idRd−1
)
. Secondly, ker
(
ϕx0− idRd−1
)
is a closed abelian normal subgroup of G:
(z, x)(v, 0)(z, x)−1 = (ϕx(v), 0) ∈ ker
(
ϕx0 − idRd−1
)
since ϕx0(ϕx(v)) = ϕx(ϕx0(v)) = ϕx(v).
In the following we consider a subgroup V ' Rn ⊂ ker
(
ϕx0 − idRd−1
)
which is normal in G
such that VZ := V∩ΛG is a lattice in V. This is the case, for example, if V is the span of a subset
of the generators e′r+1, e
′
r+2, . . . , e
′
d−1. As an immediate consequence of Lemma 2.3 we then have
Proposition 4.12. The quotient group GV := G/V is a (d−n)-dimensional almost abelian solv-
able Lie group
GV ' Rd−n−1 oϕV R,
where ϕV : R → GL(d − n − 1,R) is defined by ϕV
x[z] = (1d−1 − prV)ϕx(z) for all x ∈ R and
z ∈ Rd−1, with prV the projection of Rd−1 to V and [z] = (1d−1 − prV)(z) ∈ Rd−1/V.
We are particularly interested in the induced action of V on the twisted torus TΛG
.
Proposition 4.13. Let V, as above, be normal in G and let ΛG be normal in ΛG V. Then
the quotient map pV : TΛG
→ TΛG
/V is a principal torus bundle of rank n = dimV over an
almost abelian solvmanifold TΛ
GV
of dimension d−n. Its Chern class c1(pV) ∈ H2(TΛ
GV
,Z) can
be computed by Chern–Weil theory from the curvature of the connection κV ∈ Ω1(TΛG
,V) given
by
κV = −(prVϕ−x) · dz
in the notation of Proposition 4.12.
30 P. Aschieri and R.J. Szabo
Proof. The first statement follows from Proposition 4.3. For the Chern class, we note that the
left-invariant Maurer–Cartan one-forms on the Lie group G are given by dx and P ∈ Ω1
(
G,Rd−1
)
where
P = ϕ−x · dz,
and this descends to the twisted torus TΛG
. The desired principal Tn-connection on TΛG
is then
given by
κV = −prVP = −(prVϕ−x) · dz
and the result follows. �
By virtue of the fibration pV : TΛG
→ TΛ
GV
, the algebra of functions C(TΛG
) is also a C(TΛ
GV
)-
algebra. We are particularly interested in the case n = 1, whereby we can explicitly apply our
framework of topological T-duality. Combining Propositions 4.13 and 3.9, we immediately arrive
at
Theorem 4.14. Let y0 ∈ ker
(
ϕx0 − idRd−1
)
and let
Ry0 := R(y0, 0)
be the corresponding one-dimensional subgroup. Suppose that Ry0 is normal in G and ΛG is
normal in ΛGRy0 (this is the case, for example, if (y0, 0) is in the center of G). Let Gy0 = G/Ry0
be the almost abelian solvable Lie group constructed by Proposition 4.12, and py0 : TΛG
→ TΛGy0
the principal circle bundle constructed by Proposition 4.13. Then the C∗-algebraic T-dual
C(TΛG
) ort Ry0 ' CT
(
TΛGy0
× Ty0 , δy0
)
is a continuous-trace algebra with spectrum TΛGy0
× Ty0 and Dixmier–Douady class
δy0 = c1(py0) ^ ζy0 ,
where ζy0 is the standard generator of H1(Ty0 ,Z) ' Z and the Chern–Weil representative of
c1(py0) ∈ H2(TΛGy0
,Z) is the curvature of the connection κy0 ∈ Ω1(TΛG
) on this circle bundle
given by
κy0 = −(pry0ϕ−x) · dz.
Thus in the case of an action of R due to a normal subgroup of G that is in ker
(
ϕx0−idRd−1
)
in
the setting of Theorem 4.14, which we collectively refer to as Ry-actions, the T-dual of an almost
abelian solvmanifold TΛG
is the Hausdorff space X = TΛGy0
× Ty0 with a three-form ‘H-flux’
whose cohomology class is represented by [Hy0 ] = c1(py0) ^ ζy0 . The associated correspondence
space construction proceeds analogously to Section 3.5, which we can give explicitly as
Proposition 4.15. The topological T-duality of Theorem 4.14 is implemented by the noncom-
mutative correspondence
C
(
TΛG
× T̃y0
)
C
(
TΛG
)
[pr∗]
88
CT
(
TΛGy0
× Ty0 , δy0
)
[(py0×idT̃y0
)∗]ii
C
(
TΛGy0
) [j]
55
[p∗y0 ]
gg
as a diagram in the category RKKT.
Topological T-Duality for Twisted Tori 31
Proof. Since the subgroup
Zy0 := Ry0 ∩ ΛG
acts trivially on the algebra of functions C(TΛG
), there is an isomorphism
C(TΛG
) ort Zy0 ' C
(
TΛG
× T̃y0
)
,
where T̃y0 is the circle dual to Ty0 ' Ry0/Zy0 . Proposition 4.13 shows that the Chern–Weil
representative of c1(py0) ∈ H2
(
TΛGy0
,Z
)
pulls back to the exact two-form dκy0 ∈ Ω2(TΛG
) under
the bundle projection py0 : TΛG
→ TΛ
y0
G
. Hence p∗y0c1(py0) = 0 which implies(
py0 × idT̃y0
)∗(
c1(py0) ^ ζy0
)
= 0 ∈ H3
(
TΛG
× T̃y0 ,Z
)
.
Thus the algebra CT(TΛ
y0
G
× Ty0 , δy0) pulls back to an algebra isomorphic to C
(
TΛG
× T̃y0
)
⊗K
by py0 × idT̃y0
, and the result follows. �
4.6 Rz-actions: noncommutative torus bundles
Let us now come to the torsion summands Zmi in the Z-module decomposition (4.9). Pick
a non-trivial elementary divisor mi > 1 for some i ∈ {1, . . . , r}. The corresponding homology
generator ẽi is constructed as a Z-linear combination (4.8) of the generators (4.6). It defines
a fixed element z0 in the image of ϕx0 − idRd−1 and a corresponding one-dimensional subgroup
of G given by
Rz0 := R(z0, 0).
We further assume that
Zz0 := Rz0 ∩ ΛG
is a lattice in Rz0 . The choice of z0 ∈ Rd−1 is not unique, and any change of basis of Zd−1,
represented by a matrix B ∈ GL(d− 1,Z), defines an equally good element B · z0 ∈ Rd−1 as long
as B · z0 ∈ im
(
ϕx0 − idRd−1
)
. We write prz0 for the linear projection of Rd−1 to Rz0 , and denote
by 〈z0, z〉 ∈ R the component of z ∈ Rd−1 in Rz0 , i.e., prz0 · z = 〈z0, z〉 z0.
In the basis êi, the lattice of G is given by
ΛG = Zd−1 oϕ̂|x0 Z x0Z,
where ϕ̂x := Σ−1 ϕx Σ for all x ∈ R. Then the fibres of the underlying Mostow bundle are
‘square’ tori Td−1 with unit periodicities z ∼ z + ~ei for i = 1, . . . , d − 1, where as before ~ei
denotes the standard basis of Rd−1. The action of the subgroup Rz0 on elements (z, x) ∈ G by
right multiplication is given by
(z, x)(ζ z0, 0) =
(
z + ζΣ−1ϕx · z0, x
)
(4.10)
for ζ ∈ R, where Σ−1ϕx · z0 lies in the image of the relation matrix A = M− 1d−1.
Our principal tool to compute the C∗-algebraic T-dual for such an action of R in the image
of ϕx0 − idRd−1 , which we collectively refer to as Rz-actions, will be Green’s theorem in the
form (3.3):
C(TΛG
) ort Rz0 ∼M C0(G/Rz0) olt ΛG. (4.11)
32 P. Aschieri and R.J. Szabo
By appealing to Theorem 4.7, we may apply (4.11) fibrewise. For fixed x ∈ R, the fibre Gx of
the semi-direct product G = Rd−1oϕ̂R is the subgroup Rd−1, and the corresponding fibre of the
solvmanifold TΛG
over x ∈ R/x0 Z is the torus Td−1 = Rd−1/Zd−1. The Morita equivalence (4.11)
is C(T)-linear and the fibres of the corresponding T-dual C(T)-algebra are given by the fibrewise
Morita equivalence(
C(TΛG
) ort Rz0
)
x
' C
(
Rd−1/Zd−1
)
ortx Rz0 ∼M C0
(
Rd−1/Rz0
)
oltx Zd−1. (4.12)
The action of the subgroup Zd−1 ⊂ ΛG on the coset space Rd−1/Rz0 is induced by left multipli-
cation in the group G. After a basis transformation, we can decompose the discrete group Zd−2
into a direct sum
Zd−1 ' Zd−2
v ⊕ Zz0 ,
where Zd−1
v = (1d−1 − prz0) · Zd−1.
Let
F∗ :=
{
x ∈ R
∣∣ 〈z0,Σ
−1ϕx · z0
〉
= 0
}
.
Lemma 4.16. Over any x ∈ F∗, the fiber
(
C(TΛG
) ort Rz0
)
x
is Morita equivalent to the com-
mutative C∗-algebra C
(
Td−1
)
.
Proof. If x ∈ F∗, then w0 := Σ−1 ϕx · z0 only shifts the corresponding component of z in
(1d−1−prz0) ·Rd−1 in the Rz-action (4.10). In this case any element (z, x) ∈ G may be factorized
as
(z, x) =
(
prz0 · z + (1d−1 − prz0 − prw0
) · z , x
) (
〈w0, z〉 z0 , 0
)
.
Hence the coset space Rd−1/Rz0 ' Rd−2 can be parameterized by prz0 ·z+
(
1d−1−prz0−prw0
)
·z
with z ∈ Rd−1, which we decompose correspondingly into a direct product Rz0 ×Rd−3 (with the
second factor absent for d = 3). By (4.4) the discrete group Zd−2
v acts trivially on the line Rz0
and by translations on Rd−3, while Zz acts by translations on Rz0 and trivially on Rd−3. Then
the crossed product on the right-hand side of (4.12) may be unravelled to get
C0
(
Rd−1/Rz0
)
oltx Zd−1 ' C0
(
Rz0 × Rd−3
)
oltx
(
Zd−2
v × Zz0
)
'
[(
C0(Rz0)⊗ C0
(
Rd−3
))
oid⊗ltx Zd−2
v
]
oβx Zz0
∼M
(
C0(Rz0)⊗ C∗(Z)⊗ C
(
Td−3
))
oltx⊗id⊗id Zz0
'
(
C0(Rz0) oltx Zz0
)
⊗ C∗(Z)⊗ C
(
Td−3
)
∼M C(T)⊗ C∗(Z)⊗ C
(
Td−3
)
' C
(
Td−1
)
.
In the second line we applied Theorem 2.6. In the third line we used Example 2.14 together
with the fact that the homomorphism σZz0 from (2.5) is trivial since the groups Zz0 and Zd−2
v
are discrete, and so the action of Zz0 on the crossed product is induced by left multiplication
on Rz0 , the trivial action on Rd−3, and the trivial action on Zd−2
v . In the fifth line we used
Example 2.14 again. �
The central result of this paper is
Theorem 4.17. The C∗-algebraic T-dual of any almost abelian solvmanifold TΛG
with respect
to an Rz-action is Morita equivalent to a C∗-algebra bundle of noncommutative tori over the
circle T:
C(TΛG
) ort Rz0 ∼M
∐
x∈R/Z
Td−1
~θz0 (x)
,
Topological T-Duality for Twisted Tori 33
where the noncommutativity parameters ~θz0(x) ∈ Rd−2 are given by
~θz0(x) =
0 for x ∈ F∗,
(1d−1 − prz0)Σ−1ϕx0x · z0〈
z0,Σ−1ϕx0x · z0
〉 for x ∈ R \ F∗.
(4.13)
Proof. That the fibres over x ∈ F∗ are just ordinary tori Td−1 is established by Lemma 4.16,
so we may assume that x ∈ R \F∗. Then a simple calculation shows that any element (z, x) ∈ G
can be factorized as
(z, x) = (v, x)
(
prz0 · z〈
z0,Σ−1ϕx · z0
〉 , 0) ,
where
v :=
(
1d−1 − prz0
)
· z − 〈z0, z〉〈
z0,Σ−1ϕx · z0
〉(1d−1 − prz0
)
Σ−1ϕx · z0. (4.14)
Thus the coset space Rd−2
x,v := Rd−1/Rz0 may be parameterized by the coordinates v ∈ Rd−2 over
any x ∈ R \ F∗, and we explicitly retain the fibre index in the notation for convenience.
We now need to unravel the crossed product on the right-hand side of (4.12). From (4.4)
and (4.14) it follows that the action of elements (γv, γz0) ∈ Zd−1 ' Zd−2
v ⊕ Zz0 on the coset is
given by
(γv, 0, 0) · (v, x) = (v + γv, x), (4.15)
(0, γz0 , 0) · (v, x) =
(
v − γz0〈
z0,Σ−1ϕx · z0
〉(1d−1 − prz0
)
Σ−1ϕx · z0, x
)
. (4.16)
Applying Theorem 2.6 as in the proof of Lemma 4.16 gives
C
(
Rd−1/Zd−1
)
ortx Rz0 ∼M
(
C0
(
Rd−2
x,v
)
oltx Zd−2
v
)
oβx Zz0 ,
where the action of Zd−2
v on the coset Rd−2
x,v is given by (4.15), while the action of Zz0 on the
crossed product C0
(
Rd−2
x,v
)
oltx Zd−2
v is induced by the action on Rd−2
x,v given in (4.16) and the
trivial action on Zd−2
v .
Next, Example 2.14 yields
C0
(
Rd−2
x,v
)
oltx Zd−2
v ∼M C
(
Td−2
)
. (4.17)
Since the homomorphism σZz0 : Zz0 → R+ is trivial, it is not difficult to see that the Morita
equivalence bimoduleM implementing this equivalence is Zz0-equivariant (in the sense of Theo-
rem 2.12): the Zz0-action U : Zz0 → Aut(M) is given by Uγv(ξ)(v) = ξ(v+ γv), for all γv ∈ Zz0 ,
ξ ∈ M and v ∈ Rd−2
x,v . Applying Theorem 2.12 we conclude that the Morita equivalence (4.17)
induces the Morita equivalence(
C0
(
Rd−2
x,v
)
oltx Zd−2
v
)
oβx Zz0 ∼M C
(
Td−2
)
oltx Zz0 , (4.18)
where the action of the group Zz0 on C
(
Td−2
)
is the pullback of the action on Td−2 = Rd−2
x,v /Zd−2
v
induced from (4.16).
After rescaling x by x0 to give it unit period, this shows that the corresponding algebra of
functions on the fiber at e 2πix is that of a noncommutative d−1-torus Td−1
~θz0 (x)
with noncom-
mutativity parameter ~θz0(x) given by (4.13), see Example 2.2. Thus the C∗-algebraic T-dual
C(TΛG
) ort Rz0 is a C∗-algebra bundle of noncommutative tori A~θz0 (x)
= Td−1
~θz0 (x)
over the circle
T =
{
e 2πix |x ∈ R/Z
}
. �
34 P. Aschieri and R.J. Szabo
What becomes of the non-trivial monodromy (4.4) of the Td−1 fibers parameterized by z
in the original Mostow bundle? To answer this question, we write the monodromy matrix
M = (mij) ∈ SL(d− 1,Z) in the block form
M =
(
M|d−2 ~m
~m′ md−1 d−1
)
, (4.19)
where M|d−2 = (mij)1≤i,j≤d−2, while ~m = (mi d−1)i=1,...,d−2 and ~m′ = (md−1 i)i=1,...,d−2 are
respectively column and row vectors in Zd−2. Below we denote the usual Euclidean inner product
on Rd−2 by 〈 · , · 〉.
Proposition 4.18. The noncommutativity parameter ~θ~ed−1
(x) from (4.13) varies with a change
of coset representative x ∈ R/Z according to its SL(d − 1,Z) orbit under the action of the
monodromy matrix (4.19) by (d−2)-dimensional linear fractional transformations
~θ~ed−1
(x+ 1) = M
[
~θ~ed−1
(x)
]
:=
M|d−2 · ~θ~ed−1
(x) + ~m〈
~m′, ~θ~ed−1
(x)
〉
+md−1 d−1
. (4.20)
Under a change of basis of Zd−1 given by a matrix B ∈ GL(d−1,Z), the corresponding noncom-
mutativity parameter varies according to
~θB·~ed−1
(x+ 1) =
(
BtMB
)[
~θB·~ed−1
(x)
]
.
Proof. The key feature stems from the definition (4.2) of the monodromy matrix M:
Σ−1 ϕx+x0 = Σ−1ϕx0ϕx = MΣ−1ϕx.
The statements then follow from straightforward calculations in components. �
Remark 4.19. The right-hand side of (4.20) is an example of a linear fractional transformation
in higher dimensions, known from complex analysis, see, e.g., [12]. We can show that it defines
a Morita equivalence of noncommutative d−1-tori from Example 2.11. For this, we introduce
the skew-symmetric matrix Θ corresponding to the vector ~θ ∈ Rd−2 as in Example 2.2:
Θ =
(
0d−2
~θ
−~θ t 0
)
.
We denote by gM the element of SO(d − 1, d − 1;Z) corresponding to the monodromy matrix
M ∈ SL(d− 1,Z):
gM =
(
M 0d−1
0d−1
(
Mt
)−1
)
∈ SO(d− 1, d− 1;Z).
Introduce matrices Td−1 of order 2 with determinant −1 by
Td−1 =
(
1d−1 − Ed−1 Ed−1
Ed−1 1d−1 − Ed−1
)
∈ O(d− 1, d− 1;Z),
where Ed−1 is the matrix unit whose only non-zero element is (Ed−1)d−1 d−1 = 1. Finally, define
the element M ∈ SO(d− 1, d− 1;Z) by
M = Td−1gMTd−1.
Topological T-Duality for Twisted Tori 35
Using the adjugate formula for matrix inverses, a straightforward if tedious calculation then
shows that the corresponding SO(d − 1, d − 1;Z) orbit of Θ reproduces the (d−2)-dimensional
linear fractional transformation of (4.20):
M [Θ] =
(
0d−2 M
[
~θ
]
−M
[
~θ
]t
0
)
with M
[
~θ
]
=
M|d−2 · ~θ + ~m〈
~m′, ~θ
〉
+md−1 d−1
.
It follows that, while the fibre noncommutative tori of Theorem 4.17 are not generally identical
under a change of representative x ∈ R/Z, by Example 2.11 they are always Morita equivalent:
Td−1
~θ~ed−1
(x+1)
∼M Td−1
~θ~ed−1(x)
.
Recalling that our formulation of topological T-duality takes place in the additive category KK
from Section 3.2, this has a natural interpretation: The non-trivial monodromy in the automor-
phism group of the Td−1 fibres of the twisted torus TΛG
is manifested as a (generally non-trivial)
isomorphism in the Morita automorphism group of the fibre noncommutative tori in KK .
5 Three-dimensional solvmanifolds and their T-duals
The goal of this final section is to give some explicit examples in low dimensions of the general
formalism we have developed in Section 4, recovering some previously known results in the litera-
ture from a new perspective, as well as providing several new examples. Note that tori Td in any
dimension d are covered by our general framework of Section 4.4 for Ry-actions: when ϕx = 1d−1
for all x ∈ R, the torus bundles Td → Td−n of Proposition 4.13 are trivial. Theorem 4.14
then shows that any C∗-algebraic T-dual C
(
Td
)
ort Ry0 is isomorphic to CT
(
Td−1 × Ty0 , 0
)
'
C
(
Td−1 × Ty0
)
⊗ K. Below we apply our formalism to the well-known Mostow fibrations of
three-dimensional solvmanifolds.
5.1 Mostow bundles and SL(2,R) conjugacy classes
In three dimensions, solvmanifolds are completely classified, see, e.g., [2]. In particular, by [2,
Proposition 5.1] all solvmanifolds in this dimension which are based on connected and simply-
connected Lie groups G are almost abelian. If G = R2oϕR is abelian, i.e., ϕx = 12 for all x ∈ R,
then TΛG
' T3 is a torus, while the remaining cases correspond to non-trivial one-parameter
group actions
ϕ : R −→ SL(2,R).
Elements of SL(2,R) are classified up to conjugacy by trace, so there are three classes, de-
termined by which of the three types of conjugacy classes of SL(2,R) that the image of the
homomorphism ϕ lands in: parabolic (|Trϕx| = 2 for all x ∈ R), elliptic (|Trϕx| < 2 for all
x 6= k π/2 with k ∈ Z), or hyperbolic (|Trϕx| > 2 for all x 6= 0), see, e.g., [26].
The existence of cocompact discrete subgroups ΛG of G is highly restrictive on the allowed
homomorphisms ϕ; a necessary condition is that there exists x0 ∈ R× such that Trϕx0 ∈ Z.
The monodromy matrices
M = Σ−1 ϕx0Σ =
(
a b
c d
)
∈ SL(2,Z) (5.1)
are integer matrices that live in the corresponding conjugacy classes of SL(2,Z). As automor-
phisms of the torus fibres of the Mostow bundle, in the mapping class group SL(2,Z), they act
on T2 in the following way. Let
(
λ, λ−1
)
be the eigenvalues of M, which are the roots of the
characteristic polynomial t2 − (Tr M)t+ 1. Then there are three possibilities:
36 P. Aschieri and R.J. Szabo
� Parabolic: In this case Tr M = ± 2 and λ = λ−1 = ± 1. Then M has an integral eigenvector
corresponding to a closed curve on T2 which is invariant under the associated automor-
phism, and homeomorphisms of this type are Dehn twists of T2.
� Elliptic: In this case Tr M = 0,±1, and the eigenvalues
(
λ, λ−1
)
are complex of modulus 1.
Then the associated automorphism has finite order equal to 2, 3, 4 or 6, and corresponds
to a periodic homeomorphism of T2.
� Hyperbolic: In this case |λ| > 1 >
∣∣λ−1
∣∣. The associated automorphism has no invariant
closed curve on T2; it ‘stretches’ the eigenspace corresponding to λ and ‘contracts’ the
eigenspace corresponding to λ−1. These are called Anasov homeomorphisms of T2.
In this section we shall apply Theorem 4.17 with z0 = ~e2 = (0, 1), which for convenience we
combine with Proposition 4.18 and Remark 4.19 to reformulate it as
Theorem 5.1. The C∗-algebraic T-dual of a three-dimensional solvmanifold TΛG
with respect
to an Rz-action with z0 = (0, 1) is Morita equivalent to a C∗-algebra bundle of noncommutative
two-tori over the circle T:
C(TΛG
) ort R(0,1) ∼M
∐
x∈R/Z
T2
θ(x),
where the noncommutativity parameters θ(x) ∈ R are given by
θ(x) =
0 for x ∈ F∗,(
Σ−1ϕx0x
)
12(
Σ−1ϕx0x
)
22
for x ∈ R \ F∗,
where
(
Σ−1ϕx0x
)
ij
for i, j ∈ {1, 2} denote the matrix elements of Σ−1ϕx0x. The noncommu-
tativity parameter θ(x) varies under a change of coset representative x ∈ R/Z according to its
SL(2,Z) orbit under the corresponding monodromy matrix (5.1):
θ(x+ 1) = M
[
θ(x)
]
=
aθ(x) + b
cθ(x) + d
. (5.2)
By Example 2.10 the corresponding fibre noncommutative tori are Morita equivalent:
T2
θ(x+1) ∼M T2
θ(x),
and so isomorphic in the category KK .
The C∗-algebraic T-dual in these instances has more structure in general than the original
algebra of functions C(TΛG
), as a consequence of
Proposition 5.2. The noncommutative torus bundle
∐
x∈R/Z T2
θ(x) is a C
(
T2
)
-algebra.
Proof. We can also describe these noncommutative torus bundles via (strict) deformation quan-
tization: Performing an identical calculation to that of Example 2.1, we may describe the fibre-
wise crossed product (4.18) (with d = 3) explicitly in terms of a star-product, and hence the
C∗-algebra bundle of Theorem 5.1 as a deformation of the algebra of functions C
(
T × T2
)
on
the trivial torus bundle over T by regarding the convolution algebra Cc(T×T×Z) as the space
of functions C
(
T× T2
)
equipped with the star-product
(f ?θ g)(x, v1, v2) =
∑
(p1,p2)∈Z2
e 2πi(p1v1+p2v2)
×
∑
(q1,q2)∈Z2
fq1,q2(x)gp1−q1,p2−q2(x) e 2πiθ(x)(p1−q1)q2 , (5.3)
Topological T-Duality for Twisted Tori 37
where we used the fibrewise Fourier transformation
f(x, v1, v2) =
∑
(p1,p2)∈Z2
fp1,p2(x) e 2πi(p1v1+p2 v2),
with functions fp1,p2 : T→ C. In this formulation of the C∗-algebraic T-dual, it is immediately
evident that there is an injection C
(
T2
)
↪→
∐
x∈R/Z T2
θ(x) as the space of functions which are
independent of the coordinate v1 (or v2); for such functions, the sums in (5.3) truncate to
p1 = q1 = 0 (or p2 = q2 = 0) and the noncommutative star-product reduces to the commutative
pointwise product of functions in C
(
T2
)
. This defines C∗-algebra monomorphisms making∐
x∈R/Z T2
θ(x) into a C
(
T2
)
-algebra. �
The remainder of this paper is devoted to providing illustrations of Theorem 5.1, through
explicit calculations in each of the three conjugacy classes of SL(2,R). The features differ for
each conjugacy class so we consider them individually in turn.
5.2 Parabolic torus bundles
We start with the best studied example in the literature, which is based on the nilpotent Heisen-
berg group. In string theory it is T-dual to the three-torus T3 with H-flux by the standard
Buscher rules (see, e.g., [27]), and in topological T-duality it gives the basic example of a non-
commutative principal torus bundle [15, 21, 32]. Here we shall give a new algebraic perspective on
both these T-duals by applying our formalism of topological T-duality directly to the Heisenberg
nilmanifold. This class has several features that make it special among the three-dimensional
solvmanifolds, which we explain in detail.
Heisenberg nilmanifolds
The three-dimensional Heisenberg group Heis(3) is the nilpotent Lie group whose Mostow bundle
structure is based on the semi-direct product
Heis(3) = R2 oϕ R with ϕx =
(
1 x
0 1
)
.
Since |Trϕx| = 2, the matrix ϕx parameterizes a parabolic conjugacy class of SL(2,R). Here we
denote the coordinates on the group manifold of N = R2 by (y, z). The group multiplication
on Heis(3) is then given by
(x, y, z)(x′, y′, z′) = (x+ x′, y + y′ + xz′, z + z′),
and the inverse of a group element is
(x, y, z)−1 = (−x, xz − y,−z).
In this case it is clear that for
x0 = m
with m ∈ Z×, the matrix ϕx0 is integer-valued and hence may be taken as monodromy matrix Mm
of infinite order with Σ = 12:
Mm = ϕm =
(
1 m
0 1
)
. (5.4)
38 P. Aschieri and R.J. Szabo
The Heisenberg nilmanifold THeism(3;Z) is the compact space obtained as the quotient of Heis(3)
with respect to the lattice given by the discrete Heisenberg group
Heism(3;Z) :=
{
(mα, β, γ) ∈ Heis(3)
∣∣α, β, γ ∈ Z
}
.
The equivalence relation is given by the left action of Heism(3;Z), which leads to the local
coordinate identifications under the action of the generators of Heism(3;Z) given by
(x, y, z) 7−→ (x+m, y +mz, z),
(x, y, z) 7−→ (x, y + 1, z),
(x, y, z) 7−→ (x, y, z + 1). (5.5)
Geometrically, this exhibits the Heisenberg nilmanifold as a non-trivial principal T2-bundle
THeism(3;Z) → T. Using the algorithm described in Section 4.4, the relation matrix
Am = Mm − 12
has rank r = 1 with elementary divisors m1 = m and m2 = 0, and the first homology group
of THeism(3;Z) can thus be presented as the Z-module
H1(THeism(3;Z),Z) ' Z⊕ (Z⊕ Zm),
with respective free generators denoted ex and ey, and torsion generator ez of order m, mez = 0.
Dually, the topology of the nilmanifold is captured by its algebra of functions, which can be
computed as the subalgebra of invariant functions on the Heisenberg group Heis(3) with respect
to the left action of the lattice Heism(3;Z):
C
(
THeism(3;Z)
)
= C0
(
Heis(3)
)Heism(3;Z)
.
Harmonic analysis on the nilmanifold can be used to determine the Fourier decomposition of
any function f ∈ C(THeism(3;Z)). Invariance under the generators (5.5) of Heism(3;Z) forces the
expansion to take the form
f(x, y, z) =
∑
k∈Zm
∑
p,q∈Z
fp,k(x+mq) e 2πipy+2πi(k+mpq)z, (5.6)
for functions fp,k : R→ C vanishing at infinity. From this expression we can easily see the C(T)-
algebra structure of C(THeism(3;Z)): Restricting (5.6) to y = z = 0 determines a function on
the circle T =
{(
e 2πix/m, 1, 1
)
∈ THeism(3;Z)
}
and defines a C∗-algebra monomorphism C(T) ↪→
C(THeism(3;Z)). On the other hand, the evaluation of (5.6) at z = 0 yields a function on the
two-torus T2 =
{(
e 2πix/m, e 2πiy, 1
)
∈ THeism(3;Z)
}
and defines a C∗-algebra monomorphism
C
(
T2
)
↪→ C(THeism(3;Z)) making C(THeism(3;Z)) into a C
(
T2
)
-algebra in this case.
Following the prescriptions of Sections 4.5 and 4.6, we shall now study the known T-duals of
the Heisenberg nilmanifold through the formalism of topological T-duality discussed in Section 3.
Ry-action: T3 with H-flux
The one-parameter subgroup
ker(Am) = R(1,0) :=
{
(0, λ, 0) ∈ Heis(3)
}
is the center of Heis(3). The quotient group Heis(3)/R(1,0) can be parameterized by equivalence
classes
[
(x, 0, z)
]
, and the multiplication law[
(x, 0, z)
] [
(x′, 0, z′)
]
=
[
(x+ x′, 0, z + z′)
]
Topological T-Duality for Twisted Tori 39
is that of the abelian Lie group Heis(3)/R(1,0) ' R2. It follows that the quotient THeism(3;Z)/R(1,0)
is the two-dimensional torus T2 with coordinates
(
e 2πix/m, e 2πiz
)
. The corresponding principal
circle bundle p(1,0) : THeism(3;Z) → T2, with fibre coordinate e 2πiy, is the standard realization of
the Heisenberg nilmanifold as a T-fibration over T2 of degree m: the connection
κ(1,0) = −dy + xdz
of Theorem 4.14 has curvature
dκ(1,0) = dx ∧ dz,
and so the first Chern class c1(p(1,0)) is m times the standard generator of H2
(
T2,Z
)
' Z. By
Theorem 4.14 it follows that the C∗-algebraic T-dual
C
(
THeism(3;Z)
)
ort R(1,0) ' CT
(
T3,m
)
is a continuous-trace algebra with spectrum T3 and Dixmier–Douady class equal to m times
the standard generator of H3
(
T3,Z
)
' Z. Thus in this case we recover the standard T-duality
between the three-torus T3 with H-flux and the Heisenberg nilmanifold viewed as a circle bundle.
The correspondence space construction is given by Proposition 4.15 with TΛ
G(1,0)
= T2.
Rz-action: Noncommutative principal torus bundles
We shall now recover the known noncommutative principal torus bundle which is T-dual to
THeism(3;Z) from a new perspective, by directly working with the algebraic description of the
nilmanifold. For this, we follow our prescription for Rz-actions from Section 4.6 and consider
the one-parameter subgroup R(0,1) of Heis(3) given by
R(0,1) =
{
(0, 0, ζ) ∈ Heis(3)
}
,
whose right action by multiplication generates
(x, y, z)(0, 0, ζ) = (x, y + xζ, z + ζ).
The quotient of this action to THeism(3;Z) fixes every point of the form
(
e 2πin/m, e 2πiy, e 2πiz
)
∈
T2 ⊂ THeism(3;Z) for n ∈ Z with isotropy subgroup
Z(0,1) = R(0,1) ∩ Heism(3;Z).
In this case F∗ = ∅, and Theorem 5.1 thus identifies the C∗-algebraic T-dual through the Morita
equivalence
C
(
THeism(3;Z)
)
ort R(0,1) ∼M
∐
x∈R/Z
T2
mx
of C(T)-algebras.
We can explicitly check the anticipated monodromy (5.2): Under a change of coset rep-
resentative x ∈ R/Z of the fibres of the noncommutative torus bundle
∐
x∈R/Z T2
θm(x), the
noncommutativity parameter changes
θm(x+ 1) = θm(x) +m = Mm
[
θm(x)
]
(5.7)
by the SL(2,Z) orbit of θm(x) = mx under the monodromy matrix Mm given by (5.4), and by
Example 2.10 the fiber noncommutative tori are identical: T2
θm(x+1) = T2
θm(x). Hence in this
40 P. Aschieri and R.J. Szabo
case the non-trivial monodromy in the automorphism group of the T2 fibres of the twisted torus
is implemented trivially as an equality in its T-dual C∗-algebra bundle.
It is well known that these C∗-bundles are isomorphic to the convolution C∗-algebras of the
corresponding integer Heisenberg groups, and indeed this is precisely the original description of
these T-dual noncommutative torus bundles from [32]. For later comparison, it is instructive to
explicitly demonstrate this result using our scheme for topological T-duality, which results in
Proposition 5.3. The C∗-algebraic T-dual C
(
THeism(3;Z)
)
ort R(0,1) is Morita equivalent to the
group C∗-algebra C∗
(
Heism(3;Z)
)
.
Proof. The crux of the proof hinges on the fact that the parabolic torus bundles are the
only class in three dimensions for which a fibrewise analysis is not necessary and the Morita
isomorphisms can be implemented directly in the category KK . For this, we observe that any
element (x, y, z) ∈ Heis(3) can be uniquely factorized as
(x, y, z) = (x, y − xz, 0)(0, 0, z) =: (x, v, 0)(0, 0, z),
with the action of the subgroup R(0,1) on the normal abelian subgroup R2
(x,v) := {(x, v, 0) ∈
Heis(3)} given by
Adz(x, v, 0) := (0,0,z)(x, v, 0) = (0, 0, z)(x, v, 0)(0, 0,−z) = (x, v − xz, 0).
It follows that the Heisenberg group can alternatively be presented as the semi-direct product
Heis(3) = R2
(x,v)R(0,1). Correspondingly, its lattice is also a semi-direct product Heism(3;Z) =
Z2
(x,v) oAd Z(0,1), where Z2
(x,v) = R2
(x,v) ∩ Heism(3;Z). Thus the Morita equivalence from (3.3)
can be expressed as
C
(
THeism(3;Z)
)
ort R(0,1) ∼M C0
(
Heis(3)/R(0,1)
)
olt Heism(3;Z)
∼M C0
(
R2
(x,v)
)
olt
(
Z2
(x,v) oAd Z(0,1)
)
'
(
C0
(
R2
(x,v)
)
olt Z2
(x,v)
)
oβ Z(0,1), (5.8)
where in the last line we have further applied Theorem 2.6. Since the groups Z2
(x,v) and Z(0,1) are
discrete, the homomorphism σZ(0,1)
from (2.5) is trivial, and so the Z(0,1)-action on C0
(
R2
(x,v)
)
olt
Z2
(x,v) is the canonical action obtained from the diagonal action of Z(0,1) on Z2
(x,v) × R2
(x,v):
βζ(f)
(
(mα, ν), (x, v)
)
= f
(
(mα, ν +mαζ), (x, v + xζ)
)
for all ζ ∈ Z(0,1), (mα, ν) ∈ Z2
(x,v), (x, v) ∈ R2
(x,v) and f ∈ Cc
(
Z2
(x,v) × R2
(x,v)
)
⊂ C0
(
R2
(x,v)
)
olt
Z2
(x,v).
By Example 2.14 there is a Morita equivalence
C0
(
R2
(x,v)
)
olt Z2
(x,v) ∼M C
(
T2
)
. (5.9)
Since the homomorphism σZ(0,1)
: Z(0,1) → R+ is trivial, it follows that the Morita equivalence
bimodule M implementing this equivalence is Z(0,1)-equivariant: the Z(0,1)-action U : Z(0,1) →
Aut(M) is given by Uζ(ξ)(x, v) = ξ(x, v + xζ), for all ζ ∈ Z(0,1), ξ ∈ M and (x, v) ∈ R2
(x,v).
Applying Theorem 2.12 we conclude that the Morita equivalence (5.9) induces the Morita equiv-
alence
(
C0
(
R2
(x,v)
)
oltZ2
(x,v)
)
oβZ(0,1) ∼M C
(
T2
)
oAd∗Z(0,1), where the action of the group Z(0,1)
on C
(
T2
)
is the pullback of the action on T2 = R2
(x,v)/Z
2
(x,v) induced from the left multiplication
of Z(0,1) on Heis(3).
Topological T-Duality for Twisted Tori 41
Substituting into (5.8) we thus obtain
C(THeism(3;Z)) ort R(0,1) ∼M C
(
T2
)
oAd∗ Z(0,1). (5.10)
Now we implement Pontryagin duality via Fourier transform and apply (2.8) to the right-hand
side of (5.10) to obtain
C
(
T2
)
oAd∗ Z(0,1) ' C∗
(
Z2
(x,v)
)
o
Âd∗
Z(0,1),
where we used the fact that σ̂Z(0,1)
: Z(0,1) → R+ is trivial since Z(0,1) is discrete. Applying
Theorem 2.5, we then arrive at
C
(
THeism(3;Z)
)
ort R(0,1) ∼M C∗
(
Z2
(x,v) oAd Z(0,1)
)
= C∗
(
Heism(3;Z)
)
,
as required. �
Remark 5.4. The star-products (5.3) with θ(x) = θm(x) = mx are equivalent to the star-
products of [21, 28, 31]. In [15, 21], it is shown that the description of Proposition 5.3 defines
a noncommutative principal T2-bundle (cf. Example 4.9). It is also shown in [15] that the
monodromy (5.7), while acting trivially at the purely algebraic level, has a non-trivial action on
the K-theory group of the C∗-algebra bundle
∐
x∈R/Z T2
mx, viewed as a bundle of abelian groups
over T; a physical picture of this action in terms of monodromies of fiber D-branes is discussed
in [28].
Noncommutative correspondences
We come now to the noncommutative correspondences underlying this topological T-duality.
For this, we recall that both C
(
THeism(3;Z)
)
and C
(
T2
)
oAd∗ Z(0,1) are C
(
T2
)
-algebras, with T2
parameterized by the local coordinates (x, y). We will show that the noncommutative corre-
spondence is given by the balanced tensor product
C
(
THeism(3;Z)
)
ort Z(0,1) ' C
(
THeism(3;Z)
)
⊗C(T2)
∐
x∈R/Z
T2
mx
over C
(
T2
)
. Note that here the subgroup Z(0,1) acts non-trivially on the algebra of functions
C(THeism(3;Z)), in contrast to our previous examples. We consider elements of the convolution
algebra Cc(THeism(3;Z) × Z(0,1)) as sequences f = {fq̃}q̃∈Z(0,1)
of functions fq̃ : THeism(3;Z) → C
with the convolution product
(f ? g)q̃(x, y, z) =
∑
p̃∈Z
fp̃(x, y, z)gq̃−p̃(x, y − p̃mx, z).
We write the Fourier transformation
f(x, y, z, z̃) :=
∑
q̃∈Z
fq̃(x, y, z) e 2πiq̃z̃
for functions on THeism(3;Z) × T̃z, and define the star-product
(f ? g)(x, y, z, z̃) =
∑
q̃∈Z
(f ? g)q̃(x, y, z) e 2πiq̃z̃.
We further use the harmonic expansion of functions on the nilmanifold given from (5.6) by
fq̃(x, y, z) =
∑
k∈Zm
∑
p,q∈Z
fq̃;p,k(x+ q) e 2πiqy+2πi(k+mpq)z.
42 P. Aschieri and R.J. Szabo
After some elementary manipulations, we can then write the star-product on the space of func-
tions C
(
THeism(3;Z) × T̃z
)
as
(f ? g)(x, y, z, z̃) =
∑
l∈Zm
∑
r,s,p̃∈Z
e 2πiry+2πi(l+mrs)z e 2πip̃z̃
×
∑
k∈Zm
∑
p,q,q̃∈Z
fq̃;p,k(x+ q)gp̃−q̃;r−p,l−k(x+ q − s) e 2πimr(q−s)z e−2πimx(r−p)q̃. (5.11)
We can define an injection C
(
THeism(3;Z)
)
↪→ C
(
THeism(3;Z) × T̃z
)
as the space of functions
which are independent of the coordinate z̃. For these functions the sums in (5.11) truncate to
p̃ = q̃ = 0, and one recovers the pointwise product of functions in C
(
THeism(3;Z)
)
. On the other
hand, we can include the noncommutative algebra
∐
x∈R/Z T2
mx ↪→ C
(
THeism(3;Z) × T̃z
)
as the
space of functions which are independent of the coordinate z, and one sees that (5.11) truncates
for l = k = 0 and s = q = 0 to the star-product (5.3) with θ(x) = θm(x) = mx. Altogether, the
noncommutative correspondences are induced by the diagram
C
(
THeism(3;Z)
)
⊗
C
(
T2
) ∐
x∈R/Z
T2
mx
C
(
THeism(3;Z)
)
55
∐
x∈R/Z
T2
mx
ii
C
(
T2
)
55jj
where all arrows are ∗-monomorphisms of C∗-algebras.
5.3 Elliptic torus bundles
We shall now illustrate Theorem 5.1 for the three-dimensional solvmanifolds based on the Eu-
clidean group in two dimensions, which have no classical Hausdorff T-duals and whose noncom-
mutative T-dual fibration was first discussed in [28] for the case of the Z4 elliptic monodromy.
Here we shall recover this noncommutative geometry rigorously from our algebraic framework,
and moreover extend it to the Z2 and Z6 elliptic monodromies which have not been considered
previously.
The Euclidean group ISO(2) in two dimensions is the three-dimensional almost abelian solv-
able Lie group
ISO(2) = R2 oϕ R with ϕx =
(
cosx sinx
− sinx cosx
)
.
Since |Trϕx| < 2 for x /∈ πZ, the rotations ϕx parameterize an elliptic conjugacy class of SL(2,R).
Here we shall denote coordinates on the group manifold of N = R2 by z = (z1, z2). The group
multiplication on ISO(2) is then
(x, z1, z2) (x′, z′1, z
′
2) =
(
x+ x′, z1 + z′1 cosx+ z′2 sinx, z2 − z′1 sinx+ z′2 cosx
)
and the inverse of a group element is
(x, z1, z2)−1 =
(
−x,−z1 cosx+ z2 sinx,−z1 sinx− z2 cosx
)
.
The existence of lattices in ISO(2) is highly restrictive [26, 28]. They require angles x0 ∈ R×
for which 2 cosx0 ∈ Z, i.e., cosx0 = 0,±1
2 ,±1. For our purposes, there are three inequivalent
Topological T-Duality for Twisted Tori 43
choices of non-trivial elliptic monodromies, which are characterized by matrices M ∈ SL(2,Z)
of finite order in the cyclic subgroups Z2, Z4 and Z6. They lead to three types of Euclidean
solvmanifolds, according to the order of the monodromy in the fibre of the associated Mostow
bundle. A crucial distinction from the case of the Heisenberg nilmanifold is that here the Mostow
fibrations are not principal T2-bundles. We consider each of the three cases separately in turn.
Euclidean Z2-solvmanifolds
We observe that for
x0 = mπ
with m ∈ Z×, the matrix ϕx0 is integer-valued, and so may be set equal to M with Σ = 12.
For m even, the monodromy M = 12 is trivial and the corresponding Mostow bundle is simply
the three-torus T3, which we have already considered in Section 3.4. For m odd, which we now
assume, the nontrivial monodromy matrix is given by
M(1) = ϕmπ = −12. (5.12)
The Euclidean Z2-solvmanimanifold T
ISO
(1)
m (2;Z)
is the compact space obtained as the quotient
of ISO(2) with respect to the lattice given by the discrete Euclidean group
ISO(1)
m (2;Z) :=
{
(mπα, γ1, γ2) ∈ ISO(2)
∣∣α ∈ Z, γ = (γ1, γ2) ∈ Z2
}
.
The quotient is taken by the left action of ISO(1)
m (2;Z), which leads to the local coordinate
identifications under the action of the generators of ISO(1)
m (2;Z) given by
(x, z1, z2) 7−→ (x+mπ,−z1,−z2),
(x, z1, z2) 7−→ (x, z1 + 1, z2),
(x, z1, z2) 7−→ (x, z1, z2 + 1).
The relation matrix A(1) = M(1) − 12 = −212 has maximal rank r = 2 with elementary divisors
m1 = m2 = 2, and the first homology group of T
ISO
(1)
m (2;Z)
can thus be presented as the Z-module
H1
(
T
ISO
(1)
m (2;Z)
,Z
)
' Z⊕ (Z2 ⊕ Z2).
It follows that the Z2-solvmanifolds do not possess any classical T-duals. By symmetry the two
possible noncommutative torus bundles corresponding to the two torsion generators ez1 and ez2
of order two are the same.
Consider the one-parameter subgroup R(0,1) of ISO(2) given by
R(0,1) =
{
(0, 0, ζ) ∈ ISO(2)
}
,
which acts on the Euclidean group ISO(2) by right multiplication
(x, z1, z2)(0, 0, ζ) = (x, z1 + ζ sinx, z2 + ζ cosx).
In this case F∗ is the subset of x ∈ R where cosx = 0, so that
F∗ = π
2 (2Z + 1). (5.13)
Applying Theorem 5.1, it follows that the C∗-algebraic T-dual of the Euclidean Z2-solvmanifold
is Morita equivalent to the C(T)-algebra
C
(
T
ISO
(1)
m (2;Z)
)
ort R(0,1) ∼M
∐
x∈R/Z
T2
θ
(1)
m (x)
, (5.14)
44 P. Aschieri and R.J. Szabo
where
θ(1)m (x) =
{
0 for x ∈ 1
m
(
Z + 1
2
)
,
tan(mπx) for x ∈ R \ 1
m
(
Z + 1
2
)
.
(5.15)
Recalling that the integer m is odd here, it is easily seen that the noncommutativity para-
meter (5.15) is invariant under changing coset representative x ∈ R/Z, which is consistent with
its SL(2,Z) orbit under the corresponding monodromy matrix (5.12):
θ(1)m (x+ 1) = θ(1)m (x) = M(1)
[
θ(1)m (x)
]
,
so that fibrewise A
θ
(1)
m (x+1)
= A
θ
(1)
m (x)
. Thus the non-trivial monodromy in the automorphism
group of the T2 fibres of the twisted torus T
ISO
(1)
m (2;Z)
becomes a trivial identity action on its T-
dual C∗-algebra bundle. This is analogous to what we found in the parabolic case. Moreover, by
Proposition 5.2 the C∗-algebra bundle (5.14) can be described as a deformation of the algebra of
functions C
(
T×T2
)
on a trivial T2-bundle over the circle via a star-product f?
θ
(1)
m
g given by (5.3).
In marked contrast to the parabolic case, however, the algebra of functions C
(
T
ISO
(1)
m (2;Z)
)
is not
itself a C
(
T2
)
-algebra, and the noncommutative torus bundle cannot be identified with the
group C∗-algebra of the integer Euclidean group ISO(1)
m (2;Z).
Euclidean Z4-solvmanifolds
We next observe that for
x0 =
mπ
2
with m an odd integer, the matrix ϕx0 is again integer-valued, and so may be set equal to M
with Σ = 12:
M(2) = ϕmπ
2
= ±
(
0 1
−1 0
)
. (5.16)
Without loss of generality, we shall fix the positive sign by assuming that m ∈ 4Z+1 (the choice
of negative sign for m ∈ 4Z + 3 simply corresponds to a reflection (z1, z2) 7→ (−z1,−z2) below).
The Euclidean Z4-solvmanifold T
ISO
(2)
m (2;Z)
is the compact space obtained as the quotient of
ISO(2) with
ISO(2)
m (2;Z) :=
{(
mπ
2 α, γ1, γ2
)
∈ ISO(2)
∣∣α ∈ Z, γ = (γ1, γ2) ∈ Z2
}
.
The quotient is taken by the left action of ISO(2)
m (2;Z), which leads to the local coordinate
identifications under the action of the generators of ISO(2)
m (2;Z) given by
(x, z1, z2) 7−→
(
x+ mπ
2 , z2,−z1
)
,
(x, z1, z2) 7−→ (x, z1 + 1, z2),
(x, z1, z2) 7−→ (x, z1, z2 + 1).
The relation matrix A(2) = M(2) − 12 has maximal rank r = 2 with elementary divisors m1 = 1
and m2 = 2, and the first homology group of T
ISO
(2)
m (2;Z)
can thus be presented as the Z-module
H1
(
T
ISO
(2)
m (2;Z)
,Z
)
' Z⊕ Z2,
where the torsion generator of order two is ez2 = ez1 . Thus again there is no classical T-dual,
as is well-known in this case (see, e.g., [28]).
Topological T-Duality for Twisted Tori 45
Proceeding as in the previous case, the subset F∗ ⊂ R is again given by (5.13) and we arrive
at the Morita equivalence
C
(
T
ISO
(2)
m (2;Z)
)
ort R(0,1) ∼M
∐
x∈R/Z
T2
θ
(2)
m (x)
,
where
θ(2)m (x) =
{
0 for x ∈ 1
m(2Z + 1),
tan
(
mπ
2 x
)
for x ∈ R \ 1
m(2Z + 1).
(5.17)
The corresponding star-product f ?
θ
(2)
m
g on C
(
T × T2
)
is given by (5.3) and was originally
written down in [28], where it was also pointed out that the noncommutative torus fibration is
not isomorphic to the group C∗-algebra of ISO(2)
m (2;Z).
Compared to the previous cases, the new feature here is that a change of coset representa-
tive x ∈ R/Z generally has a non-trivial action on the fibers of the C∗-bundle
∐
x∈R/Z Aθ(2)m (x)
according to (5.2). We check this explicitly here: Recalling that m ∈ 4Z + 1 in this case, the
fibre over any x ∈ 1
mZ is preserved as then the form of (5.17) implies
θ(2)m (x+ 1) = 0 = θ(2)m (x) for x ∈ 1
mZ.
However, over any x ∈ R \ 1
mZ the noncommutativity parameter (5.17) changes non-trivally
according to the trigonometric identity tan
(
x+ mπ
2
)
= − cotx, but in a way which is consistent
with its SL(2,Z) orbit under the corresponding monodromy matrix (5.16):
θ(2)m (x+ 1) = − 1
θ(2)m (x)
= M(2)
[
θ(2)m (x)
]
for x ∈ R \ 1
mZ.
By Theorem 5.1, the fibre noncommutative tori are Morita equivalent, and so are isomorphic in
the category KK . A physical picture of the action of the monodromy on the K-theory group
of this C∗-bundle in terms of fiber D-branes is given in [28].
Euclidean Z6-solvmanifolds
Finally, we observe that for
x0 =
mπ
3
with m /∈ 3Z, the matrix ϕx0 takes four possible forms
ϕmπ
3
=
ε
2
(
ε′
√
3
−
√
3 ε′
)
with independent signs ε, ε′ = ±1. It conjugates to an integer matrix via the element Σ ∈
SL(2,R) given by
Σ =
√
2√
3
(
1 1
2
0
√
3
2
)
. (5.18)
The precise form of ϕx0 and hence of the corresponding monodromy matrix M depends on the
congruence class of the integer m in Z3:
Σ−1 ϕmπ
3
Σ =
{
±
(
1 1
−1 0
)
, m ∈ 3Z + 1,
±
(
0 1
−1 −1
)
, m ∈ 3Z + 2.
46 P. Aschieri and R.J. Szabo
Without loss of generality, we shall fix m ∈ 6Z + 2 and hence take
M(3) =
(
0 1
−1 −1
)
(5.19)
in the following (with the other three possibilities obtained simply by reflection (z1, z2) 7→
(−z1,−z2) and interchange (z1, z2) 7→ (z2, z1) below).
The Euclidean Z6-solvmanifold T
ISO
(3)
m (2;Z)
is the compact space obtained as the quotient of
ISO(2) with respect to the lattice given by the discrete subgroup
ISO(3)
m (2;Z) :=
{(
mπ
3 α,
√
2√
3
γ1 +
√
1
2
√
3
γ2,
√√
3
2 γ2
)
∈ ISO(2)
∣∣∣α ∈ Z, γ = (γ1, γ2) ∈ Z2
}
.
The quotient is taken by the left action of ISO(3)
m (2;Z), which leads to the local coordinate
identifications under the action of the generators of ISO(3)
m (2;Z) given by
(x, z1, z2) 7−→
(
x+ mπ
3 , 1
2
(√
3z2 − z1
)
,−1
2
(√
3z1 + z2
))
,
(x, z1, z2) 7−→
(
x, z1 +
√
2√
3
, z2
)
,
(x, z1, z2) 7−→
(
x, z1 +
√
1
2
√
3
, z2 +
√√
3
2
)
.
The relation matrix A(3) = M(3) − 12 has maximal rank r = 2 with elementary divisors m1 = 1
and m2 = 3, and the first homology group can thus be presented as the Z-module
H1
(
T
ISO
(3)
m (2;Z)
,Z
)
' Z⊕ Z3,
where the torsion generator of order three is ez2 = ez1 . Here too there is no classical T-dual.
Applying Theorem 5.1 in this case, with the non-trivial period matrix Σ from (5.18) and the
subset F∗ ⊂ R again given by (5.13), gives the Morita equivalence
C
(
T
ISO
(3)
m (2;Z)
)
ort R(0,1) ∼M
∐
x∈R/Z
T2
θ
(3)
m (x)
,
where
θ(3)m (x) =
{
0 for x ∈ 1
m
(
3Z + 3
2
)
,
−1
2 +
√
3
2 tan
(
mπ
3 x
)
for x ∈ R \ 1
m
(
3Z + 3
2
)
.
(5.20)
As in the previous case, changing coset representative x ∈ R/Z generally has a non-trivial
action on the fibers of the C∗-bundle over T: Recalling that m ∈ 6Z+ 2 here, the fiber over any
x ∈ 1
m
(
3Z + 3
2
)
is preserved by the form of (5.20):
θ(3)m (x+ 1) = 0 = θ(3)m (x) for x ∈ 1
m
(
3Z + 3
2
)
.
On the other hand, over any x ∈ R\ 1
m
(
3Z+ 3
2
)
the noncommutativity parameter (5.20) changes
according to the trigonometric identity
tan
(
x+ mπ
3
)
=
tanx−
√
3
1 +
√
3 tanx
,
but consistently with its SL(2,Z) orbit under the corresponding monodromy matrix (5.19):
θ(3)m (x+ 1) = − 1
θ(3)m (x) + 1
= M(3)
[
θ(3)m (x)
]
for x ∈ R \ 1
m
(
3Z + 3
2
)
.
By Theorem 5.1 the corresponding fibre noncommutative tori are Morita equivalent, and so
isomorphic in the category KK .
Topological T-Duality for Twisted Tori 47
5.4 Hyperbolic torus bundles
Our last application of Theorem 5.1 is to the final class of three-dimensional solvmanifolds,
which are hyperbolic analogues of the Euclidean solvmanifolds based on the Poincaré group,
and have also not been previously studied in the present context.
The Poincaré group ISO(1, 1) in two dimensions is the three-dimensional almost abelian
solvable Lie group which can be presented as
ISO(1, 1) = R2 oϕ R with ϕx =
(
coshx sinhx
sinhx coshx
)
.
Since |Trϕx| > 2 for all x ∈ R×, the hyperbolic rotation ϕx parameterizes a hyperbolic conjugacy
class of SL(2,R). Again we denote coordinates on the group manifold of N = R2 by z = (z1, z2),
so that the group multiplication on ISO(1, 1) is given by
(x, z1, z2)(x′, z′1, z
′
2) =
(
x+ x′, z1 + z′1 coshx+ z′2 sinhx, z2 + z′1 sinhx+ z′2 coshx
)
and the inverse of a group element is
(x, z1, z2)−1 =
(
−x,−z1 coshx+ z2 sinhx, z1 sinhx− z2 coshx
)
.
Clearly there is no x0 ∈ R× for which ϕx0 is an integer matrix in this case. However, there
is an infinite family of discrete points
x0 = log
(
m
2
±
√(m
2
)2
− 1
)
(5.21)
labelled by an integer m > 2 with 2 coshx0 = m, at which the matrix ϕx0 takes the form
ϕx0 =
1
2
(
m ±
√
m2 − 4
±
√
m2 − 4 m
)
.
This matrix has eigenvalues
(
λ, λ−1
)
where
λ±1 =
1
2
(
m±
√
m2 − 4
)
= e x0 ,
and it conjugates to the integer matrix
Mm = Σ−1ϕx0Σ =
(
m 1
−1 0
)
(5.22)
by the element Σ ∈ SL(2,R) given by
Σ =
λ±1
2
(
λ− λ−1
) − λ∓1 λ
2
(
λ2 − 1
) − 1
λ±1
2
(
λ− λ−1
) + λ∓1 λ
2
(
λ2 − 1
) + 1
. (5.23)
For definiteness, we choose the positive square root in (5.21) (with the choice of negative square
root obtained from the interchange (z1, z2) 7→ (z2, z1) below).
The Poincaré solvmanifold TISOm(1,1;Z) is the compact space obtained as the quotient of
ISO(1, 1) by the lattice
ISOm(1, 1;Z) :=
{
(x0 α,Σ · γ) ∈ ISO(1, 1)
∣∣α ∈ Z, γ = (γ1, γ2) ∈ Z2
}
.
48 P. Aschieri and R.J. Szabo
The quotient by the left action of this lattice leads to the local coordinate identifications under
the action of the generators given by
(x, z1, z2) 7−→
(
x+ cosh−1
(
m
2
)
, 1
2
(
mz1 +
√
m2 − 4z2
)
, 1
2
(√
m2 − 4z1 +mz2
))
,
(x, z1, z2) 7−→
(
x, z1 + λ2
2(λ2−1)
− λ−1, z2 + λ2
2(λ2−1)
+ λ−1
)
,
(x, z1, z2) 7−→
(
x, z1 + λ
2(λ2−1)
− 1, z2 + λ
2(λ2−1)
+ 1
)
.
The relation matrix Am = Mm − 12 has maximal rank r = 2 with elementary divisors m1 = 1
and m2 = m− 2, so the first homology group of the Poincaré solvmanifold may be presented as
the Z-module
H1
(
TISOm(1,1;Z),Z
)
' Z⊕ Zm−2,
with torsion generator ez2 = −ez1 of order m− 2.
Consider the one-parameter subgroup
R(0,1) =
{
(0, 0, ζ) ∈ ISO(1, 1)
}
acting on the Poincaré group ISO(1, 1) by right multiplication
(x, z1, z2)(0, 0, ζ) = (x, z1 + ζ sinhx, z2 + ζ coshx).
In this case the set F∗ consists of a distinguished point x∗ ∈ R on the base of the C∗-algebra
bundle given by
x∗ = tanh−1
(
2
(
λ2 − 1
)
− λ3
2
(
λ2 − 1
)
+ λ3
)
.
Applying Theorem 5.1, with the periods (5.21) and (5.23), then identifies the Morita equivalence
C
(
TISOm(1,1;Z)
)
ort R(0,1) ∼M
∐
x∈R/Z
T2
θm(x)
of C(T)-algebras, where
θm(x∗) = 0
and
θm(x) =
2λ
(
λ2 − 1
)
− λ2 + λ
(
2
(
λ2 − 1
)
+ λ
)
tanh
(
cosh−1
(
m
2
)
x
)
λ3 − 2
(
λ2 − 1
)
−
(
λ3 + 2
(
λ2 − 1
))
tanh
(
cosh−1(m2 )x
) (5.24)
for x 6= x∗.
This noncommutative torus bundle, along with its topological T-duality with C
(
TISOm(1,1;Z)
)
,
has formally the same properties as the C(T)-algebras described in Section 5.3, so we refrain
from repeating the details here. We only mention the fibre monodromy behaviour anticipated
from (5.2): Using the hyperbolic identity
tanh
(
x+ cosh−1
(
m
2
))
=
m tanhx+
√
m2 − 4
m+
√
m2 − 4 tanhx
,
we find
θm(x∗ + 1) = 0 = θm(x∗)
Topological T-Duality for Twisted Tori 49
and we see explicitly here that the noncommutativity parameter (5.24) changes consistently with
its SL(2,Z) orbit under the monodromy matrix (5.22):
θm(x+ 1) = − 1
θm(x)
−m = Mm
[
θm(x)
]
for x 6= x∗.
Thus the fiber noncommutative tori are Morita equivalent by Theorem 5.1, and so are isomorphic
C∗-algebras in the category KK .
Acknowledgments
We thank Ryszard Nest and Erik Plauschinn for helpful discussions. We thank the anonymous
referees for their detailed suggestions. This research was supported by funds from Università del
Piemonte Orientale (UPO). P.A. acknowledges partial support from INFN, CSN4, and Iniziativa
Specifica GSS. P.A. is affiliated to INdAM-GNFM. R.J.S. acknowledges a Visiting Professorship
through UPO Internationalization Funds. R.J.S. also acknowledges the Arnold–Regge Centre
for the visit, and INFN. The work of R.J.S. was supported in part by the Consolidated Grant
ST/P000363/1 from the UK Science and Technology Facilities Council.
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Topological T-Duality for Twisted Tori 51
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1 Introduction
1.1 Background
1.2 Summary and outline
2 Crossed products and duality
2.1 Dynamical systems and their crossed products
2.2 Semi-direct products and group algebras
2.3 Pontryagin duality and Fourier transform
2.4 Morita equivalence and Green's theorem
3 Topological T-duality and twisted tori
3.1 Twisted tori and their T-duals
3.2 T-duality in the category KK
3.3 Computational tools
3.4 Topological T-duality for the torus
3.5 Topological T-duality for orbifolds
4 Topological T-duality for almost abelian solvmanifolds
4.1 Mostow bundles
4.2 Almost abelian solvmanifolds
4.3 C*-algebra bundles
4.4 Rn-actions on Mostow bundles
4.5 Ry-actions: Circle bundles with H-flux
4.6 Rz-actions: noncommutative torus bundles
5 Three-dimensional solvmanifolds and their T-duals
5.1 Mostow bundles and SL(2,R) conjugacy classes
5.2 Parabolic torus bundles
5.3 Elliptic torus bundles
5.4 Hyperbolic torus bundles
References
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| id | nasplib_isofts_kiev_ua-123456789-211176 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T10:20:00Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Aschieri, Paolo Szabo, Richard J. 2025-12-25T13:22:52Z 2021 Topological T-Duality for Twisted Tori. Paolo Aschieri and Richard J. Szabo. SIGMA 17 (2021), 012, 51 pages 1815-0659 2020 Mathematics Subject Classification: 46L55; 81T30; 16D90 arXiv:2006.10048 https://nasplib.isofts.kiev.ua/handle/123456789/211176 https://doi.org/10.3842/SIGMA.2021.012 We apply the *-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 *-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 *-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 *-algebras rigorously describe the T-folds from non-geometric string theory. We thank Ryszard Nest and Erik Plauschinn for helpful discussions. We thank the anonymous referees for their detailed suggestions. This research was supported by funds from Università del Piemonte Orientale (UPO). P.A. acknowledges partial support from INFN, CSN4, and Iniziativa Speci ca GSS. P.A. is affiliated with INdAM-GNFM. R.J.S. acknowledges a Visiting Professorship through UPO Internationalization Funds. R.J.S. also acknowledges the ArnoldRegge Centre for the visit and INFN. The work of R.J.S. was supported in part by the Consolidated Grant ST/P000363/1 from the UK Science and Technology Facilities Council. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Topological T-Duality for Twisted Tori Article published earlier |
| spellingShingle | Topological T-Duality for Twisted Tori Aschieri, Paolo Szabo, Richard J. |
| title | Topological T-Duality for Twisted Tori |
| title_full | Topological T-Duality for Twisted Tori |
| title_fullStr | Topological T-Duality for Twisted Tori |
| title_full_unstemmed | Topological T-Duality for Twisted Tori |
| title_short | Topological T-Duality for Twisted Tori |
| title_sort | topological t-duality for twisted tori |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211176 |
| work_keys_str_mv | AT aschieripaolo topologicaltdualityfortwistedtori AT szaborichardj topologicaltdualityfortwistedtori |