#### Patching DFT, T-duality and gerbes

Received: February
Patching DFT, T-duality and gerbes
P.S. Howe 0
G. Papadopoulos 0
Strand 0
London WC 0
R 0
LS 0
U.K. 0
Open Access 0
c The Authors. 0
0 Department of Mathematics, King's College London
We clarify the role of the dual coordinates as described from the perspectives of the Buscher T-duality rules and Double Field Theory. We show that the T-duality angular dual coordinates cannot be identified with Double Field Theory dual coordinates in any of the proposals that have been made in the literature for patching the doubled spaces. In particular, we show with explicit examples that the T-duality angular dual coordinates can have non-trivial transition functions over a spacetime and that their identification with the Double Field Theory dual coordinates is in conflict with proposals in which the latter remain inert under the patching of the B-field. We then demonstrate that the Double Field Theory coordinates can be identified with some C-space coordinates and that the T-dual spaces of a spacetime are subspaces of the gerbe in C-space. The construction provides a description of both the local O(d, d) symmetry and the T-dual spaces of spacetime.
String Duality; Differential and Algebraic Geometry; Superstrings and Het-
1 Introduction
2 T-duality rules and patching
The T-dual circle topologically twists over the spacetime
A patching approach to T-duality
2.4 T-duality on T
3 Double field theory finite transformations B-dependent patching for dual coordinates B-independent patching for dual coordinates 2.1
4 Patching
5 A new proposal
6 Conclusions
C-spaces and DFT coordinates
Gerbes and Buscher rules
5.3 Summary of the proposal 1 4 4
One of the requirements for the consistent formulation of double and exceptional field
theories is a description of the patching conditions of doubled and exceptional spaces that
underpin these theories. Let us for simplicity focus on doubled spaces as many more results
for these are known. Doubled spaces arise by adding to the spacetime coordinates x a set
of dual coordinates x˜. In double field theory (DFT), the new coordinates are as many as
those of the spacetime.
The question that arises is how these new coordinates patch. There are two main
approaches in the literature for this. In the first approach, it is proposed that the dual
coordinates x˜ patching transformations depend on the transition functions of the B-field.
There are various suggestions for such dependence. Two such suggestions can be found
in [1, 2] and [3].
Another approach, advocated in [4], asserts that the patching conditions of the dual
coordinates x˜ can be arranged such that they do not depend on the transition functions
of the B-field. In such a case, the doubled space of any string background spacetime M
is either a product space M × Q, where Q can be chosen as Rn or T n, or the cotangent
are inert under B-field gauge transformations.
In addition, a recent proposal for DFT for some coset spaces was made in [5], following
on from an analysis of DFT for Wess-Zumino-Witten models presented in [6, 7].
One of the difficulties in deciding the way that the dual coordinates should patch is
the uncertainty of which criteria one should apply. A selection of such criteria is as follows:
• The doubled spaces patch in such a way that is consistent with the dual spaces
obtained via the Buscher T-duality rules.
quantisation property of the 3-form flux.
• The patching of double spaces is such that it requires for consistency the Dirac
• The doubled spaces satisfy the topological geometrisation condition.
• Doubled spaces can be constructed for all backgrounds with 3-form flux.
• Generalised geometry emerges naturally on doubled spaces.
The first criterion is perhaps the most conservative one. Whatever the patching of
doubled spaces is, it should reproduce both locally and globally the results that arise after
applying the Buscher T-duality rules. After all these produce the only explicit examples we
know. Locally this is indeed the case through the use of O(d, d) duality transformations [8–
11] on the fields. However, we shall see that globally the patching conditions of the doubled
spaces do not reproduce the results obtained from Buscher rules.
Moreover, it is worth mentioning that DFT has raised the expectations of what can
be described. As the transformations of DFT make no mention of isometries that are
instrumental in the Buscher rules, there is some expectation that the doubled spaces can
be used to describe a dual space which arises after dualising all spacetime directions.
Another aspect of the dualisation of the whole spacetime is the idea of geometrisation,
i.e. the notion that the theory which includes the spacetime metric and the 3-form field
strength can be described in terms of metric data only. This is analogous to Kaluza-Klein
theory which provides a geometrisation for a 2-form field strength.
The second criterion is an extrapolation of a similar result that arises in Kaluza-Klein
theory. The construction of the Kaluza-Klein space is achieved after restricting the 2-form
field strength to represent the first Chern class of a line bundle. In turn the flux of the
2-form is required to obey the Dirac quantisation condition.
The third criterion is also posed in analogy with the Kaluza-Klein theory. It states
that the pull-back of the 3-form field strength on whatever a consistent description of
doubled space is, or a generalisation of it, must represent the trivial cohomology class [13].
This has several consequences such as, for example, that the dual coordinates must have a
non-trivial topology and non-trivial transition functions over the spacetime.
The fourth criterion is a natural one from the point of view of DFT. In all proposals
made in the literature for the theory, there is no restriction mentioned on the backgrounds.
The fifth criterion is introduced because in generalised geometry the T-duality group
O(d, d) arises naturally as the (sub)group of automorphisms of a vector bundle. So the
expectation is that in a consistent formulation of the doubled space this bundle should arise
naturally. In fact it is expected to be related to, if not identified with, its tangent bundle.
There are several proposals in the literature on how the doubled spaces might patch and
some analysis of how they measure against the criteria mentioned above. In particular,
the patching of doubled spaces under the transformations proposed in [1, 2] has been
investigated in [12] where it was shown that consistency on 4-overlaps requires that the
3-form field strength H must be exact. To resolve the patching issue, C-spaces, essentially
local descriptions of gerbes, have been proposed in [13]. They exhibit consistent patching
with a cohomologically non-trivial H and locally contain the doubled spaces, but generically
they have more coordinates than doubled spaces. Indeed, in the case of non-trivial H-fields
they do not have well-defined global dimensionalities.
More recently, two new proposals for patching doubled spaces have been put forward [3,
4]. In this paper, we shall consider these two proposals and investigate them in the light of
the criteria mentioned above. First we shall clarify some aspects of the patching conditions
proposed in [3] and demonstrate that, up to an allowed redefinition of the dual coordinates
and choice of transition functions for B at double overlaps, the patching conditions of the
dual coordinates do not depend on the transition functions of the B-field. As a result, for
these choices, the dual coordinates of the doubled space remain inert under patching which
in turn implies that this proposal is related to that of [4].
The proposal made in [4] states that the dual coordinates of a doubled space remain
inert under patching and the transformations induced by the form part of a generalised
vector acting infinitesimally with a generalised Lie derivative on the fields are not coordinate
transformations but rather gauge transformations of the B-field. As a result the dual
coordinates can be forgetful and the spacetime geometry is described by a generalised
geometry structure and a splitting of the generalised geometry bundle induced by the B
field interpreted as a gerbe connection.
In the proposal of [5] for DFT on group manifolds, the doubled space is a group
manifold with the physical space embedded into it as a Lagrangian type of submanifold, after
the strong section condition is imposed, while the T-dual space corresponds to a different
embedding. In this case both the physical and dual coordinates are non-trivially patched.
In what follows we give a detailed analysis of the T-duality pair of S
of H-charge and the lens space LN = S3/ZN with 1 unit of H charge. We show that
3
the dual circle twists topologically non-trivially over the spacetime L3N and therefore that
either DFT dual coordinates cannot be identified1 with the T-duality angular coordinates,
or that the doubled spaces patching proposed in [3] and [4] is not consistent globally with
the T-duality rules. We also generalise this to other T-dual pairs including an example of
T-dual spaces constructed from the 3-torus with H-flux background.
Note that a conflict between T-duality and the strong section condition in doubled
spaces had been pointed out before from a different perspective in [14, 15]. There a
resolution was proposed by allowing additional transformations which preserve the split signature
metric on the doubled space but do not satisfy the strong section condition.
1As our results are topological, this rules out all continuous and even homotopic identifications.
We then go on to propose a scenario based on C-spaces and the Hitchin-Chatterjee
definition of a gerbe in which both the local O(d, d) symmetry and the Buscher T-dual
spaces can be consistently described. We propose an identification of the DFT
coordinate x˜ of [4], which transforms as a 1-form, with a coordinate that arises in the C-space
construction [13]. We then demonstrate how the T-dual space M˜ of a spacetime M with
H-flux and which is a circle fibration can be identified as a subspace of the total space of
the gerbe associated to H on M . We also provide explicit examples of this which include
the description of the T-dual lens space L3N = S3/ZN of S
3 with N units of H-charge as a
subspace of the total space of a gerbe on S3. The latter can be described as the union of S3
with a circle bundle with first Chern class N over an open neighbourhood of the equatorial
S2 of S3. The L3N subspace of the gerbe is the restriction of this circle bundle over the
2 of S3. We also give a similar construction for a T-dual space associated
3torus background with H-flux. As the angular coordinates that arise naturally in the gerbe
construction, and which are required for the identification of the T-dual spaces of spacetime
as subspaces of gerbes, are not included in doubled spaces and therefore not in DFT, we
conclude that, for the consistent description of a theory with manifest Buscher T-duality
symmetry, additional coordinates are required in addition to those of doubled spaces.
The paper is organised as follows: in section 2, we give the necessary and sufficient
conditions for the T-dual circle to (topologically) twist over a spacetime in a manner
consistent with the Buscher rules. We also prove that the dual circle of the lens space LN ,
viewed as a circle fibration over S2, and that of T
3 with H-flux, topologically twist over the
spacetime. In section 3, we review the proposals for patching DFT that have appeared in
the literature and in section 4 we investigate them from a patching point of view concluding
that they do not describe the topological twist of the dual circles. In section 5, we explore
the relation between doubled spaces and C-spaces, explain how local O(d, d) symmetry
arises, and present a gerbe construction for all spacetimes which are circle fibrations and
have some H-flux which allows for the identification of the T-dual space as a subspace of
the gerbe. We also present explicit examples based on S3 and T
3 with H-flux backgrounds.
In section 6, we present our conclusions.
T-duality rules and patching
To describe the Buscher T-duality rules one assumes that the spacetime M admits an S
group action generated by a vector field X which leaves the common sector fields, the
X = ∂∂θ , the metric and 2-form gauge potential can be written as
B = (dθ + qidxi) ∧ pjdxj + 2 bijdxi ∧ dxj . (2.1)
After performing a T-duality transformation, the dual metric, 2-form gauge potential and
ds˜2 = V −2(dθ˜ + pidxi)2 + gij dxidxj , B˜ = (dθ˜ + pidxi) ∧ qj dxj +
2 bij dxi ∧ dxj ,
geometry, can also have different topology to that of M . Furthermore M˜ again admits a S˜1
over the original spacetime M (or vice versa). These are given by some of the transition
functions of the B-field. As can be seen from (2.1), pi will transform under a B-field
metric to remain invariant.
The original spacetime M together with its dual M˜ can be put together to construct
respectively, with p = pidxi and similarly for q.
an enhanced space. To see this observe that the space of orbits of the S
with fixed points, let us assume from now on that the action of S1 on both spaces is free.2
In such a case, one can construct a torus bundle P (Q, T 2) over Q. The torus bundles are
M˜ viewed as circle bundles over Q. In [17], P (Q, T 2) is referred to as the correspondence
We therefore have the diagram
We can also define two-forms F and F˜ on Q by integrating H˜ over S˜1 and H over S
respectively. Here, from the T-duality rules,
2Otherwise, one can use the slice theorem to remove the fixed points and repeat the same analysis on
where h = db. So
F = −
Equations (2.4) and (2.5), together with the fact that both M and M˜ have the same
quotient Q as circle bundles, were specified as the required conditions for the two spaces
to be T-dual in [18].
The T-dual circle topologically twists over the spacetime
necessarily mean that they are (topologically) twisted over the spacetime M . To settle this
question, let us examine an example in detail. This is the well-known T-dual pair of S
with N-units of H flux and the 3-dimensional lens space L3N with 1-unit of H charge. It is
useful to note that L3N is the space of orbits of ZN on S
of ZN acts as vr → gvr, where vr are complex numbers such that v1v¯1 + v2v¯2 = 1.
3 and L3N are circle fibrations over S2, Q = S2. Moreover the first
generator of H2(S2, Z). Furthermore the cohomology groups of S3 and L3N are
H0(S3, Z) = H3(S3, Z) = Z ,
H0(L3N , Z) = H3(L3N , Z) = Z ,
H1(S3, Z) = H2(S3, Z) = 0 ,
H1(L3N , Z) = 0 ,
H2(L3N , Z) = ZN .
Next consider the T
c1(P ) = N u. In fact P =
2 fibration P = P (T 2, S2) with first Chern classes c1(P ) = u and
3 /ZN , where now the generator g of Z
computed and can be found that
H0(P, Z) = H1(P, Z) = H3(S3, Z) = H4(P, Z) = Z , H2(P, Z) = 0 .
In particular observe that the middle cohomology of P vanishes.
Consider first P as a circle fibration over S . This fibration is obtained after considering
the group action [a, vr] → [az, zvr], where z ∈ S
freely. As H2(S3, Z) = 0, all circle bundles over S
3 are topologically trivial. As a result P
1 ⊂ C, |z| = 1 is the group element and
not twist over the spacetime S .
However the T-dual Lens space L3N can also be considered as the spacetime, and so S
can be thought as its T-dual. Note that H2(L3N , Z) = Z
non-trivial circle bundles. The fibration of P over L3N is constructed by considering the
circle action [a, vr] → [az, vr]. If P was a trivial topological product S × LN , the Ku¨nneth
formula for computing the cohomology of the topological product of two spaces would have
H2(P, Z) = H2(L3N , H0(S1, Z)) = H2(L3N , Z) = ZN .
This is a contradiction as the second cohomology of P vanishes (2.7). Therefore P is
a topologically twisted product of S
non-trivial patching conditions over the spacetime LN .
1 and LN . As a result, the dual θ coordinate has
3
Incidentally, observe that P satisfies a partial version of the topological geometrisation
condition of [13]. Both the S
3 backgrounds and its dual L3N have non-trivial H fluxes. As
a result, the T-duality operation does not geometrise all of the B-flux, so that one does
not expect that the pull back of H or H˜ on P will represent the trivial class in H3(P, Z).
Instead the topological geometrisation condition manifests itself as follows: pulling back H
and H˜ onto P , one may have expected that these represent two independent cohomology
classes in H3(P, Z), but this is not the case. H3(P, Z) has one generator and the linear
pull-back operations. This is because part of the information of the transitions functions
of H and H˜ is stored in the patching conditions of P .
The example we have given above can be generalised to include T n actions and thus
T-duality in more than one direction. However, for the purpose of this paper, the example
we have investigated will suffice.
To conclude, the Buscher T-duality rules allow for the possibility that the dual circle
has a non-trivial topological twist over the spacetime, so that the dual angular coordinates
can have non-trivial patching conditions over the spacetime. As we have seen, this situation
does indeed arise in explicit examples.
A patching approach to T-duality
To give a bit more insight into the construction of circle bundle over a space and its relation
to the T-dual pairs, let us first describe how the third cohomology group of the spacetime
is constructed from the cohomology of S
1 and that of Q.3 Assuming again that S
freely on the spacetime M and that Q is simply connected, one can use the method of
spectral sequences to determine H3(M, Z) from H1(S1, Z), H2(Q, Z) and H3(Q, Z). The
construction is rather intuitive. The elements of H3(M, Z) either are generated by au,
where a is the generator of H1(S1, Z) and u are generators of H2(Q), or they are
pulledback from elements in H3(Q, Z) with the projection map. This is precisely the case if
by au may represent elements in H3(M, Z). In either case, the 3-form field strength H in
where the pull-back operation on v has been suppressed.
It is clear from the T-duality rules stated in (2.1) and (2.2) that the component of H
that take an active part in the T-duality transformations is represented by aw. Assuming
applications, let us assume that w is represented by a 2-form F˜2. The construction of M˜
3In this subsection we allow Q to have more than two dimensions in the general discussion.
where C is the 1-form gauge potential, a0 are the transition functions on double overlaps
Making use of the above, we can state the criterion for whether the dual angular
transition functions over the spacetime iff w represents a non-trivial class in H2(M, Z),
represents a non-trivial class in H2(M˜ , Z). The classes w and w˜ are represented by the
forms F˜ and F in (2.5) respectively.
T-duality on T 3 with flux
We can use the results of the previous section to demonstrate that the T-dual angular
co3 with flux also is twisted over the spacetime. For this denote the angular
co3 with (ψ1, ψ2, ψ3), 0 ≤ ψi < 2π, i = 1, 2, 3. The metric and flux are given as
H = −
been explained in the previous section, the dual coordinate topologically twists over the
spacetime iff the pull-back of dp represents a non-trivial cohomology class. Indeed
p =
dp =
generators of H2(T 3, Z).
and its pull-back on T 3 is a non-trivial class as 21π dψ2 ∧ dψ3 represents one of the three
Double field theory finite transformations
There has been extensive work in the literature to determine the allowed finite
transformations of DFT. A concise description of all possibilities and the sources can be found
in [4]. Here after imposing the strong section condition, we shall briefly summarise the
finite transformations proposed as well as their induced action on the B-field. This will
suffice for the purpose of the analysis that follows below.
First let us begin with the proposal of [1, 2]. In this proposal, the doubled space
coordinates (xi, x˜i) transform as
x′i = x′i(xj ) ,
x˜′i = x˜i − vi(x) ,
and the induced transformation on the B field is
Bi′j (x′) =
∂xk ∂xl
∂x′i ∂x′j
∂vk
∂xl
21 ∂∂xx′ki ∂∂xvjk −
∂xk ∂vi
∂x′j ∂xk
Observe that the spacetime coordinates transform with the usual diffeomorphisms while
the dual coordinates transform with a shift whose parameter depends only on the spacetime
coordinates. A modification of this proposal in the context of DFT was suggested in [16];
however, the transformations given in [16] reduce to the above after the strong section
condition has been imposed.
Another proposal for the finite transformation of DFT was put forward in [3]. For this,
a closed 2-form was introduced b, db = 0 which transforms as
while B := B − b is taken to transform tensorially:
b′ij (x′) =
∂xk ∂xl
∂x′i ∂x′j ((bkl + ∂kvl − ∂lvk )(x)),
B′ij (x′) =
∂xk ∂xl
∂x′i ∂x′j Bkl(x) .
This implies that the B-field transforms as
Bi′j (x′) =
∂xk ∂xl
∂x′i ∂x′j (Bkl + ∂kvl − ∂lvk)(x)) ,
i.e. in the same way as b. In these equations v depends only on the spacetime coordinates.
The doubled space coordinate transformations are taken to be
x′i = x′i(xj ) ,
x˜′i = x˜i + vi(x) .
x′i = x′i(xj ) ,
x˜′i = x˜i ,
More recently a new proposal has been put forward [4]. The doubled space coordinates
i.e. the spacetime coordinates transform with diffeomorphisms while the dual coordinates
remain inert with respect to B-field gauge transformations.
The B-field transforms as
i.e. in the same way as in [3].
Bi′j (x′) =
∂xk ∂xl
∂x′i ∂x′j
Bkl(x) + ∂kvl − ∂lvk (x) ,
The reason that the dual coordinates x˜ do not transform under the B-field gauge
transformations is because the component v˜j of the generalised infinitesimal vector,
V M =
that enters in the generalised Lie derivative acting on the fields, is identified as the
parameter of an infinitesimal gauge transformation of the B field viewed as a gerbe connection.
In other words v˜j is viewed as (the parameter of) a gauge transformation rather than as
a coordinate transformation. Moreover the gerbe connection introduces a splitting in the
short exact sequence
0 → T M → E → T M → 0 ,
and forms. The calculation of how this can be done has been described explicitly in [4]
and amounts to going from W generalised tensors to Wˆ ones in the notation of [4]. This
is related to the notion of the B-transform in generalised geometry [20–22]. As the dual
coordinates of the doubled space x˜ do not transform, or just transform as 1-forms, they are
inert under B-field gauge transformations. It has been argued in [4] that to describe DFT it
is sufficient to consider the diffeomorphisms of the spacetime together with the generalised
geometry structure described above which includes a splitting of the exact sequence that
determines the B field.
Let us now turn to investigate the implications of patching doubled spaces with the
transformations proposed in the previous section on the topology and geometry of spacetime.
Before we do this, let us describe a few properties of the de Rham-Cˇ ech theory as applied
that on 4-fold overlaps
d(a0βγδ − aαγδ + aαβδ − aαβγ ) = 0 .
0 0 0
If nαβγδ ∈ 2πZ, then H represents a class in H3(M, Z). The left-hand sides of all but
−Bα + Bβ; (δa)αβγ = aαβ + aβγ + aγα, and so on, and squares to zero, δ2 = 0.
We emphasise that the 2-form gauge potential B as well as the transition functions
a1, a0 are not unique in the above decomposition. In fact the decomposition is invariant
under the local “gauge” transformations
Bα → Bα + du1α , aαβ → a1αβ − u1α + u1β + dfα0β , aαβγ → aαβγ + fβ0γ − fα0γ + fα0β , (4.3)
1 0 0
where u1 are 1-forms and f 0 are functions defined on the indicated overlaps.
B-dependent patching for dual coordinates
If the coordinates for the doubled space, x˜, are taken to be one-forms patched together
using the B-field transformations, i.e.
by a1αβ = (δu1)αβ by the δ-Poincar´e lemma. This in turn implies that Bα can be shifted by
of doubled spaces is not compatible with backgrounds with non-trivial H-flux in H3(M, Z).
There are many examples of such backgrounds, for example those discussed in section 2.
Another patching proposal is that of [3] where it is asserted that the polarisation b,
with db = 0, is defined on each patch Uα of a good cover {Uα}α∈I and patches as4
has H = dB = dB, as db = 0, and so H is exact.
One can reach the same conclusion by viewing the (3.5) as a patching condition on a
∂∂xxiαkβ ∂∂xxjαlβ (Bβ − bβ)kl + (bβ)ij + (dvαβ)ij
However, as we have already mentioned the definition of B is ambiguous up to a gauge
transformation generated by u. As a result B can be chosen to be a globally defined 2-form
leading to an exact H. This result is independent from the way that the dual coordinates
transform and so it is not affected by the gauge transformation introduced in [19].
An alternative reading of the proposal made in [3], which is more tuned to the examples
described later in that paper, is as follows. One introduces two different 2-form gauge
potentials B and B for the 3-form field strength H, but where now B is no longer necessarily
tensorial. If the transition functions with respect to B and B are denoted by a1 and a0,
and a1 and a0 in the Cˇech-de Rham decomposition, respectively, we take the patching
on each Uαβ. This is similar to (4.4) and implies that aˆ1αβ = (δu1)αβ. So if we redefine
b by bα → bα − u1α on each patch b will be globally defined, while if we also redefine the
under b (or B)-field gauge transformations. This is similar to the first case discussed above,
but now does not require that the flux of H be trivial. So this interpretation leads to a
patching condition which is equivalent to one which is independent of the B-field patching.
B-independent patching for dual coordinates
Such a proposal is that described in [4]. The patching conditions are just the
diffeomorphisms of the spacetime and the patching conditions of the generalised geometry bundle
E together with a choice of a splitting. The main point is that the patching conditions of
the dual coordinates are
i.e. they remain inert. As the generalised geometry data are by construction globally
defined, the patching of such a doubled space is consistent.
the patching results that are a consequences of the Buscher T-duality rules. As we have
demonstrated with an explicit calculation in section 2, a T-dual circle can topologically
twist over the spacetime. As this cannot happen to the DFT dual coordinates, one can only
conclude that according to this proposal DFT either does not incorporate the Buscher
Tduality rules or the DFT dual coordinates x˜ should not be identified with the Buscher dual
that the DFT dual coordinates x˜ is not the full story and additional coordinates must be
introduced. There has been such a suggestion before in [13] where the basis of generalised
Wˆ generalised tensors has been identified and where it was shown how the generalised
geometry emerges. In this case, one might argue that the motivation for the introduction
of the DFT dual coordinates in the first place is somewhat weakened, or that they have
only an auxiliary status.
To enforce the idea that a generalised geometry approach is not sufficient to describe the
T-duality rules, observe that, although the generalised geometry bundle E is twisted over
the spacetime, as a space it is contractible to the spacetime M . In other words the spacetime
is fixed and the bundle transformations, which one might wish to identify with T-duality
transformations, cannot change the topology of the underlying space. On the other hand,
we have seen that T-duality changes the topology of spacetime, for example the sphere and
the lens space have different cohomology groups, and moreover both spaces in the dual pair
are smooth. This does not mean that the T-duality transformation is necessarily smooth,
but a smooth transformation of E can never induce the T-dual geometry on the spacetime,
i.e. only singular gauge transformations of E may be of interest as they may produce the
desirable T-dual space.
The modified proposal of [3] discussed above also suffers a similar problem in that the
patching condition (4.8) does not reproduce the Buscher rules and cannot accommodate
dual angular coordinates.
A new proposal
C-spaces and DFT coordinates
Here we shall propose a scenario which illustrates the role of the various coordinates and
how the Buscher T-dual spaces can be incorporated using the C-space construction of [13].
= 0
nates at double intersections are associated with the fibre directions of the principal U(1)
bundles that arise in the Hitchin-Chatterjee description of gerbes [23, 24], explained in
Common sector theories with O(d, d) local gauge symmetry can be described solely in
terms of generalised geometry, i.e. without the introduction of additional coordinates. Such
theories can also be described in terms of C-spaces, as discussed in [13]. In this context of
C-spaces additional one-form coordinates can be introduced, as we have seen above, and it
was shown in [13] that the first patching condition in (5.1) can be used to introduce new
subordinate to the good cover. It seems reasonable on the grounds of their transformation
properties to identify the y˜
incorporates the the DFT doubled coordinates into a C-space description.
However, we have shown that the Buscher T-dual spaces cannot be described in terms
of the (x, x˜) coordinates alone. So the question that remains is where the Buscher T-dual
spaces are hidden in this description. The C-space description contains in addition the
Buscher T-dual spaces are hidden in the gerbe.
Although in the construction of C-spaces a good cover has been used, for the definition of
a Hitchin-Chatterjee gerbe any open cover5 suffices. We shall use this to adapt an open
cover such that the Buscher T-duals can be described as subspaces of gerbes.
To illustrate how gerbes can be constructed, consider the example of S
of H flux. We have already seen that the T-dual space of this is the lens space L3N with
one unit of flux. To describe this gerbe on S3 [25], we can choose a stereographic cover of
open interval, and the Mayer-Vietoris description of H3(S3, Z) which uses representatives
localised on U01, see e.g. [26]. Such representatives are constructed as follows. As U01 is
{U0, U1}. Observe that at the intersection
there are no more than double overlaps the rest of the compatibility conditions for the gerbe
are trivially satisfied. The gerbe6 associated to S3 and H is then the union of S3 together
with the principal U(1) bundle on U01 which has first Chern class N u. Observe that the
principal bundle over U01 when restricted on S
significant that the lens space L3N which is the T-dual to S
3 naturally appears in this gerbe
with the fibre coordinate of the lens space that appears in the gerbe construction. We shall
provide a further explanation for this below.
the pull back operation from Q to M . Choose a cover on S
to U01 by F01. Choose the gerbe principal U(1) bundle P01 on U01 to have Chern class
represented by F . Then a representative of the class of the 3-form flux H on M can be
constructed as in equation (5.4)
defined on M and it is a representative of the 3-form flux associated to the gerbe. The
5Note, however, that for gerbes there is a notion of refinement [23]. As a result, any chosen open cover
can be refined to a good open cover, so that any gerbe can be related to one defined on a good open cover.
The gerbe in not a manifold. From the perspective of S3 it grows an extra dimension as one approaches
the sphere at the equator.
T-dual space of M is the bundle space of P01 which is clearly a subspace of the total space
as in section 2.4 in which the T-dual pair of T 3 with flux was described. In particular, we set
F = dp =
the T-dual space T˜3 as described by the Buscher T-duality rules. For a different treatment
2 ⊂ U01 ⊂ T 3 is
of this example, see [5].
As a final example we take M to be a circle bundle over Q with 3-form flux H that can
→ Q is the projection and ϕα : π−1(Wα) → Wα × S1 is
that {ϕα−1(Wα × Vr)}, r = 0, 1, is a cover for M . As the union of open sets is open U0 =
Sα ϕα−1(Wα ×V0) and U1 = Sα ϕα−1(Wα ×V1) are open and cover M . As in the case that M
a partition of unity subordinate to the cover {U0, U1}. Hˆ is globally defined and represents
circle represent the generator of H1(S1, Z). It is clear that the gerbe is the union of
spacetime with a circle bundle defined on the open set U01 of M which is the restriction of the
pull-back of a circle bundle over the base space Q with Chern class w. The circle bundle over
Q is the T-dual space derived from the Buscher rules. If S3 is viewed as a circle fibration
over S2 and H represents N units of flux, the above gerbe construction will also lead to the
identification of the T-dual space as LN . It is clear that the gerbes in all the above examples
have simple descriptions because the spacetimes have been covered by only two open sets.
Summary of the proposal
The above results provide evidence to suggest that the double coordinates x˜ of DFT that
transform like 1-forms [4] should be identified with the y˜ coordinates that occur in C-spaces,
eqn (5.3). DFT can be formulated with only these coordinates and will exhibit local O(d, d)
symmetry as such a description accommodates generalised geometry both from the double
spaces point of view and that of C-spaces. However, such a formulation will not describe
the T-dual spaces of the spacetime. This is regardless of the choice of solution to the strong
section condition that one makes on the doubled space.
Our results have also established that the T-dual space of a spacetime with H flux can
be identified as a subspace of a gerbe which is part of the C-space. This has been done
explicitly for the T-dual space derived after performing T-duality along the fibre direction
of a spacetime which is a circle fibre bundle. This identification requires the presence of
additional coordinates from those of doubled space which are the fibre coordinates of the
principal U(1) bundles that lie on double intersections of an open cover of the spacetime
and are required in the description of the gerbe.
We have given two gerbe descriptions of the T-dual space of S
flux example. The first description was in terms of a stereographic cover and the other in
terms of a cover adapted to the fibration over S2. In both cases, the T-dual space has been
identified as the lens space LN . In hindsight this may have been expected. The T-dual
space should be independent from a large enough selection of covers on the spacetime that
are used to describe the gerbe. This can be seen as the requirement for the construction of
gerbes and that of T-dual spaces to be covariant. In turn one can view this as a covariant
description of the Buscher T-duality rules.
We have made a proposal based on C-spaces and the Hitchin-Chatterjee description of a
gerbe where both the local (bundle) O(d, d) symmetry and the T-dual spaces of a spacetime
can be described in a globally consistent way. In particular, we have demonstrated that the
doubled space of a DFT as described in [4] can be included into a C-space and the O(d, d)
symmetry arises as part of the generalised geometry structure on C-spaces. Furthermore,
we have demonstrated that the T-dual spaces of a spacetime that are constructed using
Buscher rules can be identified as subspaces of the gerbe which is included in C-spaces
but not in the doubled spaces. In this identification, the T-dual angular coordinate of a
spacetime which is a circle fibration with T-duality operation taken along the fibre circle
is identified with the gerbe angular coordinate which is the fibre coordinate of a principal
U(1) bundle defined on an intersection of two open sets of the spacetime.
Our analysis has indicated that it is not possible to formulate a theory which exhibits
both local O(d, d) symmetry and at the same time has a description of all the T-dual
spaces of a spacetime based only on doubled spaces. Using the available globally consistent
definitions of doubled spaces, we have demonstrated that these cannot provide an
explanation for the property of the T-dual circles to topologically twist over the spacetime. This
topological twisting has been established in several examples and it is a consequence of the
Buscher rules. In other words, the T-dual spaces cannot arise in DFT as different solutions
to the (strong) section condition on doubled spaces.
The inclusion of gerbes in a consistent definition of a theory which exhibits local O(d, d)
symmetry and which describes the T-dual spaces of a spacetime requires the presence of
additional angular coordinates, the gerbe coordinates. Such spaces are not manifolds and in
particular they do not have a fixed dimension. Nevertheless they contain all the necessary
ingredients for the definition of the theory including the ability to perform differential
geometry computations related to O(d, d) symmetry and the topological properties required
for the descrption of the T-dual spaces.
The gerbe description of T-dual spaces of a spacetime has some additional
consequences. First notice that the Buscher rules are not covariant. Their formulation involves
several gauge choices and their construction is essentially local on the spacetime. Moreover,
they depend on the spacetime admitting an isometry. On the other hand gerbes can be
defined on any smooth manifold with a closed 3-form flux H without further additional
assumptions. Therefore the gerbe description can be seen as a covariantisation of the
Tduality rules. Furthermore the gerbe description opens the possibility that it might be
possible to investigate the T-duals of a spacetime that does not admit isometries. In this
case, however, it may not be possible to identify the subspaces of the gerbe which can be
characterised as T-dual spaces as we have done in the case of spacetimes with isometries.
Even if the T-dual spaces can be identified, it is likely that they will not be manifolds.
GP is partially supported by the STFC rolling grant ST/J002798/1.
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