Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
aa r X i v : . [ h e p - t h ] F e b Prepared for submission to Complex Manifolds,special issue: Generalized Geometry
QMUL-PH-21-11
Towards an extended/higher correspondence
Generalised geometry, bundle gerbes and global Double Field Theory
Luigi Alfonsi
Centre for Research in String Theory,School of Physics and Astronomy,Queen Mary University of London,327 Mile End Road, London E1 4NS, UK [email protected]
Abstract
In this short paper, we will review the proposal of a correspondence between the doubled geom-etry of Double Field Theory and the higher geometry of bundle gerbes. Double Field Theoryis T-duality covariant formulation of the supergravity limit of String Theory, which generalisesKaluza-Klein theory by unifying metric and Kalb-Ramond field on a doubled-dimensional space.In light of the proposed correspondence, this doubled geometry is interpreted as an atlas de-scription of the higher geometry of bundle gerbes. In this sense, Double Field Theory can beinterpreted as a field theory living on the total space of the bundle gerbe, just like Kaluza-Kleintheory is set on the total space of a principal bundle. This correspondence provides a highergeometric interpretation for para-Hermitian geometry which opens the door to its generalisationto the other Extended Field Theories.This review is based on, but not limited to, my talk at the workshop
Generalized Geometry andApplications at Universität Hamburg on 3rd of March 2020.
Keywords : bundle gerbes, para-Hermitian geometry, T-duality, generalised geometry
MSC classes : ontents double / string correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 The doubled space/bundle gerbe correspondence . . . . . . . . . . . . . . . . . . 143.3 Recovering generalised geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 The NS5-brane is a higher Kaluza-Klein monopole . . . . . . . . . . . . . . . . . 20 One of the most characteristic and fascinating features of String Theory, when compared tothe usual field theories, is the appearance of T-duality: an additional, hidden symmetry ofthe theory. Double Field Theory is a T-duality covariant formulation of the supergravity limitof String Theory which makes this symmetry manifest. Double Field Theory was proposedin [HZ09] and seminal work includes [Sie93a, Sie93b]. See [BT14, BB20] for reviews in thecontext of Extended Field Theories. As enlightened by [Ber19, BB20], Double Field Theory canbe interpreted as a generalisation of Kaluza-Klein theory, which geometrically unifies the metricwith the Kalb-Ramond field, instead of a gauge field.
The higher geometry of T-duality.
The Kalb-Ramond field is, geometrically, the con-nection of a bundle gerbe G ։ M , a categorification of a U (1) -bundle, which was introducedby [Mur96,MS00] and reformulated in terms of Čech cohomology by [Hit01]. In [NSS15], bundlegerbes are formalised as principal ∞ -bundles in the context of higher geometry. Given a goodcover { U α } of the base manifold M , the connection of a bundle gerbe is given by local -forms B ( α ) ∈ Ω ( U α ) , local -forms Λ ( αβ ) ∈ Ω ( U α ∩ U β ) and local scalars G ( αβγ ) ∈ C ∞ ( U α ∩ U β ∩ U γ ) , hich are patched on overlaps of patches by H = d B ( α ) B ( β ) − B ( α ) = dΛ ( αβ ) Λ ( αβ ) + Λ ( βγ ) + Λ ( γα ) = d G ( αβγ ) G ( αβγ ) − G ( βγδ ) + G ( γδα ) − G ( δαβ ) ∈ π Z . (1.0.1)Since the Kalb-Ramond field is the connection of a bundle gerbe, T-duality has been naturallyformulated in the context of higher geometry. Topological T-duality [BEM04b,BEM04a,BHM04,BHM05] is based on the topological properties of bundle gerbes and T-duality has been formu-lated as an isomorphism of bundle gerbes in [BN15, FSS17a, FSS17b, FSS18a, FSS18b, NW19].Let us consider two T n -bundle spacetimes M π −→ M and f M e π −→ M over a common base mani-fold M . Then the couple of bundle gerbes G Π −→ M and e G e Π −→ f M , encoding two Kalb-Ramondfields respectively on M and f M , are geometric T-dual if the following isomorphism exists G × M f M M × M e GG M × M f M e G M f MM ∼ =T - dualityΠ e π π e ΠΠ π e π e Π π e π (1.0.2)This diagram is closely related to the diagram in [CG11], which formalises T-duality in thecontext of generalised geometry: T K ⊕ T ∗ KT M ⊕ T ∗ M T f M ⊕ T ∗ f MT M ⊕ T ∗ M ⊕ M × R nπ ∗ e π ∗ /T n / e T n (1.0.3)where we called K := M × M f M . This is because we can interpret the Courant algebroid asa geometric object which embodies the infinitesimal symmetries of a bundle gerbe [Gua11]. Inthis sense T-duality is a geometric property of bundle gerbes. .1 Introduction to local Double Field Theory Here, we will give a brief introduction to the formalism of local Double Field Theory.
Doubled patch.
Let us consider an open simply connected d -dimensional open patch U .We can introduce coordinates ( x µ , e x µ ) : U → R d , which we will call collectively x M := ( x µ , e x µ ) .Now, we want to equip the vector space R d with the fundamental representation of the contin-uous T-duality group O ( d, d ) . Since the action of O ( d, d ) -matrices on R d preserves the matrix η MN := ( ) , we can define a metric η = η MN d x M ⊗ d x N ∈ ⊙ T ∗ U with signature ( d, d ) . Gauge algebra.
We want now to define a generalised Lie derivative which preserves the η -tensor, i.e. such that L X η = 0 for any vector X ∈ X ( U ) . Thus, for any couple of vectors X, Y ∈ X ( U ) we can define (cid:0) L X Y (cid:1) M := X N ∂ N Y M − P MLNP ∂ L X N Y P , (1.1.1)where we defined the tensor P MLNP := δ MP δ LN − η ML η NP , (1.1.2)which projects the GL (2 d ) -valued function ∂ L X N into an o ( d, d ) -valued one. The generalisedLie derivative is also known as D-bracket J X, Y K D := L X Y . The C-bracket is defined as theanti-symmetrisation of the D-bracket, i.e. J X, Y K C := 12 (cid:0) J X, Y K D − J Y, X K D (cid:1) . (1.1.3)Now, if we want to construct an algebra of generalised Lie derivatives, we immediately find outthat it cannot be close, i.e. we generally have (cid:2) L X , L Y (cid:3) = L J X,Y K C (1.1.4)Thus, to assure the closure, we need to impose extra conditions. The weak and the strongconstraint (also known collectively as section condition) are respectively the conditions η MN ∂ M ∂ N φ i = 0 , η MN ∂ M φ ∂ N φ = 0 (1.1.5)for any couple of fields or parameters φ , φ . The immediate solution to the section conditionis obtained by considering only fields and parameters φ which satisfy the condition e ∂ µ φ = 0 .Therefore, upon application of the strong constraint, all the fields and parameters will depend onthe d -dimensional submanifold U := U / ∼ ⊂ U , where ∼ is the relation identifying points withthe same physical coordinates ( x µ , e x µ ) ∼ ( x µ , e x ′ µ ) . In particular vectors X ∈ X ( U ) satisfying thestrong constraint can be identified with sections of the generalised tangent bundle T U ⊗ T ∗ U ofgeneralised geometry. Moreover the C-bracket, when restricted to strong constrained vectors,reduces to the Courant bracket of generalised geometry, i.e. we have J − , − K C (cid:12)(cid:12)(cid:12) e ∂ µ =0 = [ − , − ] Cou (1.1.6) n this sense, the geometry underlying Double Field Theory, when strong constrained, locallyreduces to generalised geometry. Generalised metric.
We can define the generalised metric G = G MN d x M ⊗ d x N by requir-ing that it is symmetric and it satisfies the property G ML η LP G P N = η MN . Thus, the matrix G MN can be parametrised as G MN = g µν − B µλ g λρ B ρβ B µλ g λν − g µλ B λν g µν ! . (1.1.7)where g := g µν d x µ ⊗ d x ν is a symmetric tensor and B := B µν d x µ ∧ d x ν is an anti-symmetrictensor on the submanifold U . These can be respectively interpreted as a metric and a Kalb-Ramond field on the d -dimensional patch U . If we consider a strong constrained vector V := v + e v ∈ X ( U ) ⊕ Ω ( U ) . The infinitesimal gauge transformation given by generalised Lie derivative δ G MN = L V G MN is equivalent to the following gauge transformations: δg = L v g, δB = L v B + d e v (1.1.8)where L v is the ordinary Lie derivative. This, then reproduces the gauge transformations ofmetric and Kalb-Ramond field. Therefore, the infinitesimal generalised diffeomorphisms of the d -dimensional patch U unify the infinitesimal diffeomorphisms of the d -dimensional subpatch U ⊂ U with the infinitesimal gauge transformations of the Kalb-Ramond field, in analogy withKaluza-Klein theory. The globalisation problem.
However, the Kalb-Ramond field B is geometrically theconnection of a bundle gerbe and hence it is globalised by the patching conditions (1.0.1).Thus, it is not obvious how the local patches (cid:0) U , ∼ , η, G (cid:1) , which we introduced here, can beconsistently glued together? This is the substance of the globalisation problem of the doubledgeometry underlying Double Field Theory. Seminal work in this direction was done by [BCP14].The correspondence which is the object of this paper is a candidate to answer this question.We know how to globalise the local geometry of Double Field Theory for particular classes ofexamples, where the gerby nature of the Kalb-Ramond field is not manifest. In particular, globalDouble Field Theory on group manifolds [HRE09, BHL15, BdBHL15, Has18] is well-defined.Also, doubled torus bundles [Hul07], which are globally affine T n -bundles on an undoubledbase manifold [BHM07], are well-defined. However, there is no conclusive answer on how toglobalise this geometry in the most general case. Moreover, it has been argued in [HS13] thatthe doubled torus bundles should be recoverable by imposing a certain compactified topology toa general doubled space, whose geometry, however, remains an open problem. A problem whichbecomes even more obscure in the case of the geometry underlying Exceptional Field Theory. .2 Para-Hermitian geometry for Double Field Theory The first proposal of formalisation of the geometry underlying Double Field Theory as a para-Kähler manifold was developed by [Vai12] and then generalised to a para-Hermitian manifoldby [Vai13]. The para-Hermitian program was further developed by [FRS17,Svo18,MS18,FRS19,MS19, Shi19, HLR19, BPV20, BP20, IS20, Svo20].
Para-complex geometry.
An almost para-complex manifold ( M , J ) is a d -dimensionalsmooth manifold M which is equipped with a (1 , -tensor field J ∈ End( T M ) , called almostpara-complex structure, such that J = id T M and that the ± -eigenbundles L ± ⊂ T M of J have both rank( L ± ) = d . A para-complex structure is, then, equivalently given by a splittingof the form T M = L + ⊕ L − (1.2.1)Therefore, the structure group of the tangent bundle T M of the almost para-complex manifoldis reduced to GL ( d, R ) × GL ( d, R ) ⊂ GL (2 d, R ) . The para-complex structure also canonicallydefines the following projectors to its eigenbundles: Π ± := 12 (1 ± J ) : T M − ։ L ± . (1.2.2)An almost para-complex structure J is said to be, respectively, ± -integrable if L ± is closedunder Lie bracket, i.e. if it satisfies the property (cid:2) Γ( M , L ± ) , Γ( M , L ± ) (cid:3) Lie ⊆ Γ( M , L ± ) . (1.2.3)The ± -integrability of J implies the existence a foliation F ± of the manifold M such that L ± = T F ± . An almost para-complex manifold ( M , J ) is a para-complex manifold if and onlyif J is both + -integrable and − -integrable at the same time. Para-Hermitian geometry.
An almost para-Hermitian manifold ( M , J, η ) is an almostpara-complex manifold ( M , J ) equipped with a metric η ∈ J T ∗ M of Lorentzian signature ( d, d ) which is compatible with the almost para-complex structure as it follows: η ( J − , J − ) = − η ( − , − ) . (1.2.4)A para-Hermitian structure ( J, η ) canonically defines an almost symplectic structure ω ∈ Ω ( M ) ,called fundamental -form, by ω ( − , − ) := η ( J − , − ) . An almost para-Hermitian manifold can beequivalently expressed as ( M , J, ω ) , since the para-Hermitian metric can be uniquely determinedby η ( − , − ) = ω ( J − , − ) . Notice that the subbundles L ± are both maximal isotropic subbundlesrespect to η and Lagrangian subbundles respect to ω . Recovering generalised geometry.
The para-Hermitian metric immediately induces anisomorphism η ♯ : L ± ∼ = −−→ L ∗∓ . In the case of a + -integrable para-Hermitian manifold, this mplies the existence of an isomorphism T M ∼ = T F + ⊕ T ∗ F + (1.2.5)given by X Π + ( X ) + η ♯ (Π − ( X )) , for any vector X ∈ T M . As shown by [FRS17, MS18], itis possible to define a bracket structure J − , − K D : X ( M ) × X ( M ) → X ( M ) which is compatiblewith the para-Hermitian metric, so that ( T M , J − , − K D , η ) is a metric algebroid, and whichmakes a generalised version of the Nijenhuis tensor of J vanish [MS18, p. 13]. If we consider anycouple of sections X + ξ, Y + ζ ∈ Γ( M , T F + ⊕ T ∗ F + ) , the bracket can be rewritten as J X + ξ, Y + ζ K D = (cid:0) [ X, Y ] + L X ζ − ι Y d ξ (cid:1)| {z } Dorfman bracket on T F + ⊕ T ∗ F + + (cid:0) [ ξ, ζ ] ∗ + L ∗ ξ Y − ι ζ d ∗ X (cid:1) (1.2.6)where [ − . − ] ∗ , L ∗ ( − ) and d ∗ are operators induced by the Lie bracket of T M . Therefore, if werestrict ourselves to couples strongly foliated vectors, i.e. X + ξ, Y + ζ ∈ X ( F + ) ⊕ Ω ( F + ) , werecover the usual Dorfman bracket J X + ξ, Y + ζ K D = [ X, Y ] + L X ζ − ι Y d ξ, (1.2.7)i.e. we recover generalised geometry.An almost para-Hermitian manifold ( M , J, η ) is, in particular, a para-Kähler manifold if thefundamental -form is symplectic, i.e. d ω = 0 . In the general case, the closed -form K ∈ Ω ( M ) defined by K := d ω , which embodies the obstruction of ω from being symplectic, is interpretedas the generalised fluxes of Double Field Theory. Born geometry.
A Born geometry is the datum of an almost para-Hermitian manifold ( M , J, ω ) equipped with a Riemannian metric G ∈ J T ∗ M which is compatible with both themetric η and the fundamental -form ω as it follows: η − G = G − η and ω − G = −G − ω. (1.2.8)Such a Riemannian metric can be identified with the generalised metric of Double Field Theory. Generalised T-dualities.
The generalised diffeomorphisms of Double Field Theory cannow be identified with diffeomorphisms of M which preserve the para-Hermitian metric η ,i.e isometries Iso( M , η ) . The push-forward of a generalised diffeomorphism f ∈ Iso( M , η ) isnothing but an O ( d, d ) -valued function f ∗ ∈ C ∞ ( M , O ( d, d )) . This group of symmetries can befurther extended to the group of general bundle automorphisms of T M preserving the para-Hermitian metric η . A generalised diffeomorphism induces a morphism of Born geometries ( M , J, ω, G ) ( M , f ∗ J, f ∗ ω, f ∗ G ) , (1.2.9)which is an isometry of the para-Hermitian metric, i.e. such that it preserves η = f ∗ η . articularly interesting is the case of b -shifts, which can be seen as a bundle morphisms e b : T M → T M covering the identity id M of the base manifold. This transforms the para-complexstructure by J J + b , which also implies ω ω + b . Therefore, a b -shift maps the splitting T M = L + ⊕ L − to a new one T M = L ′ + ⊕ L − , preserving the eigenbundle L − , but not L + .Therefore, it does not preserve + -integrability. The patching puzzle of para-Hermitian geometry.
However, as shown in [Alf20b], ifwe are interested in recovering a general conventional geometric background, given by a generalspacetime manifold M equipped with a general bundle gerbe connection ( B ( α ) , Λ ( αβ ) , G ( αβγ ) ) ,we encounter a conceptual problem.If we want to consider a conventional bosonic supergravity background, there must exists afoliation F − of M such that L − = T F − and the leaf space M := M / F − of this foliationmust be a smooth manifold. Thus, the foliation F − is simple and the canonical quotient map π : M ։ M = M / F − is a surjective submersion, making M a fibered manifold. Let now ( e x µ , x µ ) be local coordinates adapted to the foliation F − , i.e. fibered, on any patch U α . Then,the fundamental -form ω ∈ Ω ( M ) must have the form [MS19, p. 40] ω = d e x ( α ) µ ∧ d x µ ( α ) − π ∗ B ( α ) . (1.2.10)Since, this satisfies π ∗ H = d( π ∗ B ( α ) ) = − d ω , where H ∈ Ω ( M ) is the curvature of the Kalb-Ramond field, we would expect to be possible for the local -forms B ( α ) to be patched as ageneral connection of a bundle gerbe. In other words, by defining the patches of the leaf spaceby U α := π ( U α ) , we would expect the following general patching conditions B ( β ) − B ( α ) = dΛ ( αβ ) on U α ∩ U β , Λ ( αβ ) + Λ ( βγ ) + Λ ( γα ) = d G ( αβγ ) on U α ∩ U β ∩ U γ ,G ( αβγ ) + G ( βαδ ) + G ( γβδ ) + G ( δαγ ) ∈ π Z on U α ∩ U β ∩ U γ ∩ U δ (1.2.11)to be allowed. However, as shown by [Alf20b], the transition functions of M on two-fold overlapsof patches U α ∩ U β force, in general, the bundle gerbe to be trivial. Therefore, if we want toembed a conventional supergravity background into an almost para-Hermitian manifold, wehave some troubles.Thus, we have two main questions to answer about para-Hermitian geometry. Firstly, whydoes it work so well? We are currently unable to provide a well-defined generalisation of para-Hermitian geometry to any other Extended Field Theory, but if we could be able to derivepara-Hermitian geometry from a more fundamental geometric principle, perhaps this wouldgive us the key to find such a generalisation. Secondly, how can we modify its globalisationsuch that recovering a conventional supergravity background becomes possible? The formalismproposed in [Alf20a, Alf20b] tries to answer both questions. To explain how, we first need tointroduce more elements of higher geometry. Bundle gerbes
In this section we will briefly introduce some fundamental notions in higher geometry, withparticular focus on bundle gerbes. For an introductory self-contained review, see [Bun21].Let
Diff be the ordinary category of smooth manifolds and
Grpd ∞ the ( ∞ , -category of Lie ∞ -groupoids. A smooth stack X is defined as an ∞ -functor X : Diff op −→ Grpd ∞ (2.0.1)which satisfies some higher gluing properties, known as descent. This can be though as ageneralisation of the notion of sheaf which takes value in Lie ∞ -groupoids. We will call H the ( ∞ , -category of smooth stacks on manifolds. Example 2.1 (Manifolds as smooth stacks) . Given any smooth manifold M ∈ Diff , we caneasily construct a sheaf C ∞ ( − , M ) ∈ H of smooth functions to M , which is in particular astack. This is nothing but a Yoneda embedding Diff ֒ → H of the smooth manifolds into the ( ∞ , -category of stacks.The abelian bundle gerbe is a categorification of the principal U (1) -bundle introduced by [Mur96,MS00]. More recently, in [NSS15], the bundle gerbe has been reformalised as a special case ofprincipal ∞ -bundle, where the structure Lie -group is G = B U (1) , i.e. the circle -group. Definition 2.2 (Circle -group) . The circle -group B U (1) ∈ H is defined as the group-stackwhich sends a smooth manifold M to the groupoid B U (1)( M ) whose objects are U (1) -bundleson M and whose morphisms are bundle isomorphisms. The group-stack structure is given bythe following bundle isomorphisms P − ⊗ P ∼ = M × U (1) , P ⊗ P − ∼ = M × U (1) ,P ⊗ ( P ′ ⊗ P ′′ ) ∼ = ( P ⊗ P ′ ) ⊗ P ′′ (2.0.2)where, for any given circle bundle P , we called P − the circle bundle with opposite st Chernclass, i.e. such that c ( P − ) = − c ( P ) .Thus, the tensor product ⊗ plays the role of the group multiplication, the trivial bundle M × U (1) plays the role of the identity element and P − plays the role of the inverse element of P .Let us now give a concrete description of this geometrical object. Definition 2.3 (Bundle gerbe) . Let M be the smooth manifold that we can identify with usualspacetime. A bundle gerbe is defined as a principal B U (1) -bundle G Π −−→→ M by the followingpullback diagram in the ( ∞ , -category H of higher smooth stacks: G ∗ M B U (1) Π f (2.0.3) here the higher stack B U (1) := B ( B U (1)) is the delooping of the group-stack B U (1) and themap f : M → B U (1) is the Čech cocycle of the bundle gerbe. Remark 2.4 (Bundle gerbe in local data) . Let U := { U α } be any good cover for the basemanifold M . The Čech groupoid ˇ C ( U ) is defined as the ∞ -groupoid corresponding to thefollowing simplicial object · · · F αβγ U α ∩ U β ∩ U γ F αβ U α ∩ U β F α U α ˇ C ( U ) . (2.0.4)Now, by using the natural equivalence between the Čech groupoid ˇ C ( U ) and the manifold M inthe ( ∞ , -category of stacks, we can express the map between M and the moduli stack B U (1) as a functor of the form M ≃ ˇ C ( U ) B U (1) . f (2.0.5)By using the definition of the Čech groupoid, such a map can be presented as a collection ofcocycles F αβ U α ∩ U β → B U (1) which are glued by isomorphisms on three-fold overlaps ofpatches F αβγ U α ∩ U β ∩ U γ . Since, as we have seen, any map U → B U (1) from an open set U is equivalently a U (1) -bundle P ։ U , we obtain the following diagram: (cid:8) µ ( αβγ ) : P αβ ⊗ P βγ ∼ = −−→ P αγ (cid:9) F αβ P αβ F αβγ U α ∩ U β ∩ U γ F αβ U α ∩ U β F α U α M (2.0.6)More in detail, we have a collection of circle bundles { P αβ ։ U α ∩ U β } on each overlap ofpatches U α ∩ U β ⊂ M such that:• there exists a bundle isomorphism P αβ ∼ = P − βα on any two-fold overlap of patches U α ∩ U β ,• there exists a bundle isomorphism µ ( αβγ ) : P αβ ⊗ P βγ ∼ = −−→ P αγ on any three-fold overlapof patches U α ∩ U β ∩ U γ ,• this isomorphism satisfies µ ( αβγ ) ◦ µ − βγδ ) ◦ µ − γδα ) ◦ µ ( δαβ ) = 1 on any four-fold overlaps ofpatches U α ∩ U β ∩ U γ ∩ U δ .We, thus, recovered the Hitchin-Chatterjee formulation [Hit01] of the bundle gerbe Π : G − ։ M . Remark 2.5 (Topological classification of bundle gerbes) . Notice that the trivialisation weintroduced defines a Čech cocycle corresponding to a class (cid:2) G ( αβγ ) (cid:3) ∈ H ( M, Z ) in the rdcohomology group of the base manifold M . Thus bundle gerbes are topologically classified by aclass dd( G ) ∈ H ( M, Z ) , called Dixmier-Douady class of the bundle gerbe. Lemma 2.6 (Automorphisms of the bundle gerbe) . As seen by [Bun20], the -group of auto- orphisms of a bundle gerbe G Π −−→→ M is Aut( G ) = Diff( M ) ⋉ H ( M, B U (1)) (2.0.7)where H ( M, B U (1)) = B U (1)( M ) is the -group of U (1) -bundles with connection on M . Noticethat this is nothing but the higher geometric version of the gauge group G DFT = Diff( M ) ⋉ Ω ( M ) of DFT proposed by [Hul15]. In fact, the natural map U (1) -Bundles ( M ) → Ω ( M ) isjust the curvature map sending a U (1) -bundle to its curvature b ∈ Ω ( M ) . In this section we will construct the correspondence between the doubled geometry of DoubleField Theory and the higher geometry of bundle gerbes. We will define the atlas of a bundlegerbe and we will show that it can be identified with the doubled space of Double Field Theory.This will have the consequence that Double Field Theory can be globally interpreted as a fieldtheory on the total space of a bundle gerbe, just like ordinary Kaluza-Klein theory lives on thetotal space of a principal bundle.
Definition 3.1 ( -truncation of stacks) . Let H be the ordinary category of sheaves on mani-folds. Then, the inclusion H ֒ → H has a left adjoint τ : H → H which is called -truncationand which sends a higher stack X ∈ H to its restricted sheaf τ X ∈ H at the -degree. Definition 3.2 (Atlas of a smooth stack) . The atlas of a smooth stack X ∈ H is defined by asmooth manifold A equipped with a morphism of smooth stacks Φ :
A −→ X (3.0.1)which is, in particular, an effective epimorphism, i.e. whose -truncation τ Φ :
A − ։ τ X isan epimorphism of sheaves. See [Hei05] and [Lur06] for more detail.This formalizes the idea that to any geometric stack X ∈ H we can associate an atlas which ismade up of ordinary manifolds A . This provides a remarkably handy tool to deal with highergeometric objects. Moreover, the notion of atlas will be a pivotal in establishing a correspondencebetween doubled and higher geometry. Remark 3.3 (Gluing morphisms of stacks) . Given a geometric stack X ∈ H equipped withan atlas Φ :
A −→ X , we can write the Čech nerve of Φ as the following simplicial object . . . A × X A × X A A × X A A X . Φ (3.0.2)For simplicity, let us now consider just a geometric -stack X ∈ H . A complicated object suchas a morphism of stacks σ : X −→ S , for some S ∈ H , can be equivalently expressed on theatlas A of the stack X . This can be done as the map induced by the atlas A X S Φ σ σ (3.0.3) ogether with an isomorphism of the two maps induced by the kernel pair of the atlas A × X A S σσ ′ (3.0.4)such that it satisfies the cocycle condition on A × X A × X A . For more details, see [Hei05].The idea of gluing morphisms of stacks on the atlas will be useful in this section, when we willhave to consider geometric structures on a bundle gerbe. double / string correspondence The aim of this section will be to prove the existence of a correspondence between doubled andhigher geometry in a linearised form.
Remark 3.4 (Atlas of an L ∞ -algebra) . By linearising the notion of atlas of a smooth stack,we obtain that the atlas of an L ∞ -algebra g can be defined by an ordinary Lie algebra atlas equipped with a homomorphism of L ∞ -algebras φ : atlas − ։ g that is surjective onto the -degree truncation of g . Definition 3.5 ( string Lie -algebra) . Let us call string := R d ⊕ b u (1) the -algebra of theabelian Lie -group R d × B U (1) . It is well-understood that any L ∞ -algebra g is equivalentlydescribed in terms of its Chevalley-Eilenberg dg-algebra CE( g ) . In our particular case this is CE( string ) = R [ e a , B ] / h d e a = 0 , d B = 0 i , (3.1.1)where { e a } with a = 0 , . . . , d − are generators in degree and B is a generator in degree .The Lie -algebra string = R d ⊕ b u (1) can be interpreted as a linearisation of a bundle gerbe.In this sense, it is trivially made up of a flat Minkowski space and a trivial Kalb-Ramond field. Lemma 3.6 ( double / string correspondence) . The atlas of the Lie -algebra string is the para-Kähler vector space (cid:0) R d ⊕ ( R d ) ∗ , J, ω (cid:1) , where• J is the para-complex structure corresponding to the canonical splitting R d ⊕ ( R d ) ∗ ,• ω is the symplectic structure given by the transgression of the higher generator of string . Proof.
Recall the definition 3.4 of atlas φ : atlas − ։ string for an L ∞ -algebra. The map φ can be dually given as an embedding φ ∗ : CE( string ) ֒ −→ CE( atlas ) between their Chevalley-Eilenberg dg-algebras. Thus, we want to identify an ordinary Lie algebra atlas such that itsChevalley-Eilenberg dg-algebra contains a -degree element ω := φ ∗ ( B ) ∈ CE( atlas ) (3.1.2)which is the image of the -degree generator of CE( string ) and which must satisfy the equation d ω = 0 , (3.1.3) iven by the fact that a homomorphism of dg-algebras maps φ ∗ (0) = 0 . Clearly, this ele-ment must also be a singlet under Lorentz transformations and, thus, it must not have freeLorentz-indices. Since atlas must be an ordinary Lie algebra, its Chevalley-Eilenberg dg-algebra CE( atlas ) will only have -degree generators. Thus its generators must consist not only in theimages e a := φ ∗ ( e a ) , but also in an extra set e e a for a = 0 , . . . , d − which satisfies ω = e e a ∧ e a . (3.1.4)Now, the equation d ω = 0 , combined with the equation d e a = 0 , implies that the differential ofthe new generator is zero, i.e. d e e a = 0 . Therefore, we found the dg-algebra CE( double ) = R [ e a , e e a ] / h d e a = 0 , d e e a = 0 i (3.1.5)where we renamed the Lie algebra atlas to double . This ordinary Lie algebra is immediately double = (cid:0) R d ⊕ ( R d ) ∗ , [ − , − ] = 0 (cid:1) , i.e. the abelian Lie algebra whose underlying d -dimensionalvector space is R d ⊕ ( R d ) ∗ . Now, recall that the Chevalley-Eilenberg dg-algebra CE( g ) of anyordinary Lie algebra g is isomorphic to the dg-algebra (cid:0) Ω • li ( G ) , d (cid:1) of left-invariant differentialforms on the corresponding Lie group G = exp( g ) . Therefore, we have the isomorphism CE( double ) ∼ = (cid:0) Ω • li ( R d,d ) , d (cid:1) (3.1.6)where we called R d,d the abelian Lie group integrating double whose underlying smooth manifoldis still the linear space R d × ( R d ) ∗ . Thus, the smooth functions C ∞ ( R d,d ) are generated bycoordinate functions x a and e x a and the basis of left-invariant -forms on R d,d is simply givenby e a = d x a , e e a = d e x a (3.1.7)Thus, the transgressed element ω ∈ CE( double ) is equivalently a symplectic form ω = d e x a ∧ d x a .Moreover, the canonical splitting R d ⊕ ( R d ) ∗ induces a canonical para-complex structure J ,which is compatible with the symplectic form ω . Therefore, the atlas of string is equivalently apara-Kähler vector space (cid:0) R d ⊕ ( R d ) ∗ , J, ω (cid:1) . Remark 3.7 (Emergence of para-Hermitian geometry) . On one side of the correspondence,the Lie -algebra string = R d ⊕ b u (1) is the linearisation of a bundle gerbe and, on the otherside, the para-Kähler vector space (cid:0) R d ⊕ ( R d ) ∗ , J, ω (cid:1) is the linearisation of a para-Hermitianmanifold. The latter is the atlas of the former. Remark 3.8 (Kernel pair of the atlas of string ) . Now let us discuss the kernel pair of the atlas φ : atlas − ։ string . This is defined as the pullback (in the category theory sense) of two copiesof the map φ of the atlas. The coequalizer diagram of these maps is double × string double double string . φ (3.1.8)To deal with it, we can consider the Chevalley-Eilenberg algebras of all the involved L ∞ -algebrasand look at the equalizer diagram of the cokernel pair which is dual to the starting kernel pair tlas (cid:0) R d ⊕ ( R d ) ∗ , J, ω (cid:1) R d ⊕ b u (1) bundle gerbe para-Hermitian manifold(i.e. doubled space)Figure 1: para-Hermitian geometry (i.e. the geometry ofdoubled spaces) as the atlas description of bundle gerbes. (3.1.8). This will be given by the following maps of differential graded algebras: CE( double ) ⊔ CE( string ) CE( double ) CE( double ) CE( string ) . φ ∗ (3.1.9)Let us describe this in more detail. When composed with φ ∗ , the two maps at the centre of thediagram both send the generators e a ∈ CE( string ) to e a ∈ CE( double ) ⊔ CE( string ) CE( double ) .However, they map the generator B ∈ CE( string ) to two different elements ω = e e a ∧ e a and ω ′ = e e ′ a ∧ e a , where e e a and e e ′ a are such that they both satisfy the same equation d e e ′ a = d e e a .This implies that they are related by a gauge transformation e e ′ a = e e a + d λ a . This can be seenas a consequence of the gauge transformations B ′ = B + d λ with parameter λ := λ a e a . Remark 3.9 (T-duality on the double algebra) . The ordinary Lie algebra double is not the atlasonly of the Lie -algebra string , but of an entire class of Lie -algebras. For example, we have doublestring ^ string e φφ (3.1.10)where we called ^ string the Lie -algebra whose Chevalley-Eilenberg dg-alegbra is given by CE (cid:0) ^ string (cid:1) = R (cid:2)e e a , e B (cid:3) / h d e e a = 0 , d e B = 0 i and where e φ is the atlas mapping the genera-tors by e e a e e a and e B e a ∧ e e a . The Lie -algebra ^ string can be immediately seen as theT-dualisation of string along all the d directions of the underlying spacetime. More generally, double will be the atlas of any T-dual of the Lie -algebra string . This is nothing that a linearisedversion of T-duality of bundle gerbes. In the previous subsection, we established a correspondence between linearised doubled geome-tries and L ∞ -algebras, which interprets the former as the atlas description of the latter. In thissubsection we will globalise this relation to construct a correspondence between doubled spacesand bundle gerbes. emark 3.10 (On the nature of the extra coordinates) . The d -dimensional atlas of the bundlegerbe is the natural candidate for being an atlas for the doubled space where Double Field Theorylives. This way, we can completely avoid the conceptual issue of postulating many new extradimensions in extended geometry, because the extra coordinates which appears in the extendedcharts describe the degrees of freedom of a bundle gerbe. In this sense, a flat doubled space R d,d can be seen as a coordinate description of a trivial bundle gerbe. Remark 3.11 (Atlas for the Lie -group) . Let R d × B U (1) be the Lie -group which integratesthe Lie -algebra string := R d ⊕ b u (1) . Let us call again R d,d the the ordinary Lie group whichintegrates the ordinary abelian Lie algebra R d ⊕ ( R d ) ∗ . Therefore, we have a homomorphism ofLie groups exp( φ ) : R d,d − ։ R d × B U (1) , (3.2.1)which exponentiates the homomorphism of Lie algebras φ : atlas − ։ string from the previoussubsection. Consequently, this is also a well defined atlas for the Lie -group R d × B U (1) . Lemma 3.12 (Doubled space/bundle gerbe correspondence) . The atlas of a bundle gerbe G Π −−→→ M is a para-Hermitian manifold ( M , J, ω ) , where• J is the para-complex structure corresponding to the splitting of T M into horizontal andvertical bundle induced by the connection of the bundle gerbe,• ω is the fundamental -form given by the transgression of the connection of the bundlegerbe, i.e. which satisfies π ∗ H = − d ω with H ∈ Ω ( M ) curvature of the bundle gerbe. Proof.
Let G ։ M be a bundle gerbe on a base manifold M . Thus G can be locally trivialisedas a collection of local trivial gerbes { U α × B U (1) } α ∈ I on a given open cover { U α } α ∈ I of the basemanifold M . Thus, we have an effective epimorphism ϕ α : R d × B U (1) ։ U α × B U (1) for anychart. These can be combined in a single effective epimorphism F α ∈ I R d × B U (1) { ϕ α } α ∈ I −−−−−−−−→→ G .Thus, we can cover the bundle gerbe with copies of the Lie -group R d × B U (1) . Since thisLie -group comes equipped with the natural atlas (3.2.1), we can define the composition maps Φ α : R d,d exp( φ ) −−−−−−→→ R d × B U (1) ϕ α −−−−→→ U α × B U (1) . By combining them we can construct aneffective epimorphism Φ : G α ∈ I R d,d { Φ α } α ∈ I −−−−−−−−→→ G (3.2.2)From now on, let us call the total space of the atlas M := F α ∈ I R d,d . Notice that, in general,this is a disjoint union of R d,d -charts. We can now use the map (3.2.2) to explicitly constructthe Čech nerve of the atlas. What we obtain is the following simplicial object: G α,β,γ ∈ I R d,d × G R d,d × G R d,d G α,β ∈ I R d,d × G R d,d G α ∈ I R d,d G , { Φ α } α ∈ I which tells us how the charts of the atlas are glued by morphisms. Let us describe this diagramin more detail in terms of its dual diagram of Chevalley-Eilenberg dg-algebras. Let us also call (cid:0) B ( α ) , Λ ( αβ ) , G ( αβγ ) (cid:1) the Čech cocycle of the bundle gerbe. The two maps of the kernel pair end the local -degree generator to d x µ and the local -degree generator to a couple of local -forms ω triv( α ) = d e x ( α ) µ ∧ d x µ and ω triv( β ) = d e x ( β ) µ ∧ d x µ on the fiber product of the α -th and β -th charts. Now the local -forms d e x ( α ) µ and d e x ( β ) µ are required to be related by a gaugetransformation d e x ( α ) µ = d e x ( β ) µ + dΛ ( αβ ) µ where the gauge parameters Λ ( αβ ) µ are given bythe cocycle of the bundle gerbe. Equivalently, the two -forms must be related by a gaugetransformation ω triv( α ) = ω triv( β ) + dΛ ( αβ ) with gauge parameter Λ ( αβ ) := Λ ( αβ ) µ d x µ . The gaugeparameters are required to satisfy the cocycle condition Λ ( αβ ) + Λ ( βγ ) + Λ ( γα ) = d G ( αβγ ) onthree-fold overlaps of charts.On the atlas M of the bundle gerbe, we can define a -form ω ∈ Ω ( M ) by taking the difference ω ( α ) := ω triv( α ) − π ∗ B ( α ) of the local -form ω triv( α ) and the pullback of the local connection -form B ( α ) of the bundle gerbe from the base manifold on each chart R d,d . This definition assures that ω ( α ) = ω ( β ) on overlaps of charts R d,d × G R d,d . Therefore, this -form is globally well-definedand we can write it simply as ω , by removing the α -index. In local coordinates we can write ω = (cid:0) d e x ( α ) µ + B ( α ) µν d x ν (cid:1) ∧ d x µ (3.2.3)Notice that the form ω is, more generally, invariant under gauge transformations of the bundlegerbe. From the definition of ω , we obtain the relation with curvature of the bundle gerbe: π ∗ H = − d ω, with H ∈ Ω ( M ) . (3.2.4)Now, we want to show that M is canonically para-Hermitian with fundamental -form ω . Theprojection π : M ։ M induces a short exact sequence of vector bundles: ֒ −→ Ker( π ∗ ) ֒ −→ T M π ∗ −−−→→ π ∗ T M −−→→ (3.2.5)From the definition of the -form ω , we can see that it is a projector to the vertical bundle ω : T M − ։ Ker( π ∗ ) (3.2.6)Therefore, the -form ω defines the splitting π ∗ ⊕ ω into horizontal and vertical bundle T M ∼ = π ∗ T M ⊕ Ker( π ∗ ) . (3.2.7)This splitting canonically defines a para-complex structure J ∈ Aut( T M ) . If we split any vectorin horizontal and vertical projection X = X H + X V , the para-complex structure J is defined suchthat J ( X ) = X H − X V . Notice that, since J defines a splitting T M = L + ⊕ L − of the tangentbundle of M , as seen in section 1.2, this identifies L + ≡ π ∗ T M and L − ≡ Ker( π ∗ ) . Therefore,the atlas of a bundle gerbe is a para-Hermitian manifold ( M , J, ω ) with para-complex structure J and fundamental -form ω , defined above. Remark 3.13 (Principal connection of the bundle gerbe) . ω ∈ Ω ( G ) . Let us rewrite G ω −−→ Ω , here Ω is just a sheaf over manifolds. Thus we have M G Ω α ω ( α ) ω , M × G M Ω ω ( α ) ω ( β ) . However, Ω is a sheaf, thus the -morphism in the second diagram is just an identity. In otherwords we obtain that ω ∈ Ω ( G ) is given on the atlas by a collection of local -forms, ω ( α ) onany chart, such that ω ( α ) = ω ( β ) on any overlap of charts. Remark 3.14 (Analogy with a principal U (1) -bundle) . This way of expressing the connectionon our atlas is, despite of the appearance, very natural and familiar. When we write theconnection of a U (1) -bundle in local coordinates, we are exactly writing a -form ω ( α ) := d θ ( α ) + A ( α ) µ ( x ( α ) )d x µ ( α ) ∈ Ω ( R d +1 ) on the local chart R d +1 , where (cid:8) d θ ( α ) , d x µ ( α ) (cid:9) is the coordinatebasis of Ω ( R d +1 ) . On the overlaps of charts we have ω ( α ) = ω ( β ) , which assures that the the -form we are writing in local coordinates is equivalently the pullback ω ( α ) = φ ∗ α ω of a well-defined -form ω on the total space of the U (1) -bundle. Example 3.15 (Topologically trivial doubled space) . Let us consider a topologically trivialbundle gerbe G = M × B U (1) . The corresponding doubled space is a para-Kähler manifold ( M , J, ω ) where M = T ∗ M is just the cotangent bundle of the base manifold, the para-complexstructure J corresponds to the canonical splitting T M ∼ = T M ⊕ T ∗ M and the connection ω = d e x µ ∧ d x µ is the canonical symplectic form on T ∗ M with { x µ , e x µ } Darboux coordinates.
Example 3.16 (Doubled Minkowski space) . If, in the previous example, we choose as basemanifold the Minkowski space M = R d , the corresponding doubled space will be the para-Kähler vector space ( R d,d , J, ω ) . Remark 3.17 (Correspondence between sections of the bundle gerbe and the doubled space) . Let us consider again a topologically trivial bundle gerbe G = M × B U (1) . Any section M I ֒ −→ G will be a U (1) -bundle I ։ M , while any section M ι ֒ −→ M will be an embedding e x = e x ( x ) .These two objects are immediately related by ι ∗ ω = curv( I ) (3.2.8)where curv( − ) is the curvature -form of a U (1) -bundle. Since any bundle gerbe can be locallytrivialised, it is possible to generalise this relation to the general topologically non-trivial case.The correspondence between bundle gerbes and doubled spaces was firstly presented and studiedin [Alf20a] by using this observation. Remark 3.18 (A doubled-yet-gauged space) . The principal action of the bundle gerbe is trans-gressed to the atlas by a shift ( x µ , e x µ ) ( x µ , e x µ + λ µ ( x )) in the unphysical coordinates, whichcan be identified with a gauge transformation B B + d( λ µ d x µ ) of the Kalb-Ramond field.Moreover, the property G / B U (1) ∼ = M of bundle gerbes, when transgressed to the atlas, canbe identified with the idea that physical points correspond to gauge orbits of the doubledspace [Par13]. Remarkably, this gives a global geometric interpretation of the strong constraint f Double Field Theory [Alf20a]. Therefore, the atlas of the bundle gerbe is naturally a doubled-yet-gauged space, according to the definition given by [Par13]. Remark 3.19 (Basis of global forms) . In general it is also possible to express the principalconnection ω = e e a ∧ e a in terms of the globally defined -forms e e a = d e x ( α ) a + B ( α ) aν d x ν and e a = d x a on the atlas. We pack both in a single global -form E A with index A = 1 , . . . , d which is defined by E a := e a and E a := e e a . In this basis, we have that the connection can beexpressed by ω = ω AB E A ∧ E B , where ω AB is the d -dimensional standard symplectic matrix. Definition 3.20 (Generalised metric) . A global generalised metric can be defined, in analogywith a Riemannian metric, as an orthogonal structure G G −−→ GL (2 d ) //O (2 d ) on the bundlegerbe. On the atlas ( M , J, ω ) , this will be given by a collection of Riemannian metrics G ( α ) withthe following pacthing conditions: M G GL (2 d ) //O (2 d ) . Φ α G ( α ) G , M × G M GL (2 d ) //O (2 d ) G ( α ) G ( β ) As explained in [Alf20a], if we require the generalised metric structure to be invariant under theprincipal B U (1) -action of the bundle gerbe, this will have to be of the form G = G AB E A ⊙ E B = g ab e a ⊙ e b + g ab e e a ⊙ e e b (3.2.9)where we called the matrix G AB := ( g ⊕ g − ) AB and where g ∈ ⊙ Ω ( M ) is a Riemannian metricon the base manifold. In the coordinate basis { d x µ ( α ) , d e x ( α ) µ } we find the usual expression G ( α ) MN = g µν − B ( α ) µλ g λρ B ( α ) ρβ B ( α ) µλ g λν − g µλ B ( α ) λν g µν ! , (3.2.10)where B ( α ) is the connection of the bundle gerbe. As explained in remark 3.18, invariance underthe principal B U (1) -action can be seen as the global geometric version of the strong constraintof Double Field Theory. This was called higher cylindricity condition in [Alf20a], in analogywith Kaluza-Klein theory. Here we will show that generalised geometry is naturally recovered from the bundle gerbe per-spective upon imposition of the strong constraint, i.e. invariance under the principal B U (1) -action. Remark 3.21 (Generalised geometry on the atlas) . Let { ∂ M } = { ∂ µ , e ∂ µ } be the local coor-dinate basis of T M . A vector on the atlas ( M , J, ω ) can be written in local coordinates as V = V M ( α ) ∂ M = v µ ( α ) ∂ µ + e v ( α ) µ e ∂ µ , where the components V M ( α ) are locally defined. The fundamen-tal -form ω will project this into a vertical vector ω ( V ) = ( e v ( α ) µ + B ( α ) µν v ν ( α ) ) e ∂ µ . Now, if we all { D A } the basis of globally defined vectors on M dual to the global -forms { E A } , we canwrite a vector on the atlas by V = V A D A , where now the components V A are globally defined.We can now express the isomorphism π ∗ ⊕ ω in (3.2.7) by V A D A = v µ ( α ) ∂ µ + (cid:0)e v ( α ) µ + B ( α ) µν v ν ( α ) (cid:1)e ∂ µ (3.3.1)Notice that, if we restrict ourselves to strong constrained vectors, i.e. vectors whose components V M ( α ) only depend on the coordinates of the base manifold M , these are immediately sections ofa Courant algebroid twisted by the bundle gerbe G ։ M with local potential B ( α ) .We have already shown that strong constrained vectors on the atlas reduce to sections of aCourant algebroid. Now, we want to show that the bracket structure of the Courant algebroidalso comes from the bundle gerbe. This was mostly explored in [Alf20a]. Definition 3.22 (Tangent stack of the bundle gerbe) . We can define the tangent stack of abundle gerbe G as the internal hom stack T G := [ D , G ] , where D := Spec( R [ ǫ ] / h ǫ i ) is theinfinitesimally thickened point. Remark 3.23 (Atiyah sequence of the bundle gerbe) . A calculation shows that [ D , M ] = T M and [ D , B U (1)] = B U (1) ⋉ b u (1) , where b u (1) is the Lie -algebra of the Lie -group B U (1) .From this, we obtain the short exact sequence ֒ −→ G ⋉ b u (1) ֒ −→ T G π ∗ −−−→→ π ∗ T M −−→→ . (3.3.2)This sequence is nothing but the stack version of the short exact sequence (3.2.5). The connectionof the bundle gerbe induces the isomorphism of stacks T G ∼ = π ∗ T M ⊕ G ⋉ b u (1) , (3.3.3)which is the stack version of the isomorphism (3.2.7). Definition 3.24 (Atiyah L ∞ -algebroid of the bundle gerbe) . We can define the Atiyah L ∞ -algebroid of our bundle gerbe by at G := T G // B U (1) ։ M , in perfect analogy with the Atiyahalgebroid of a principal bundle. Remark 3.25 (Courant -algebra) . The isomorphism (3.3.3) induces the isomorphism of L ∞ -algebroids on the manifold M at G ∼ = T M ⊕ s M × b u (1) , (3.3.4)where ⊕ s is a semi-direct sum. This, on sections, gives the isomorphism of L ∞ -algebras Γ( M, at G ) ∼ = X ( M ) ⊕ s b u (1)( M ) , (3.3.5)where b u (1)( M ) is the -algebra of line u (1) -bundles on M . As shown by [Col11], the Lie -algebra structure of X ( M ) ⊕ s b u (1)( M ) is isomorphic to the Lie -algebra structure of thestandard Courant -algebra, whose -bracket is the Courant bracket [ − , − ] Cou . If we write ections V, W ∈ Γ( M, at G ) of the Atiyah L ∞ -algebroid on the atlas, in the notation of remark3.21, we will have the -bracket [ V, W ] Cou = [ v, w ] Lie + L v e w − L w e v −
12 d( ι v e w − ι w e v ) + ι v ι w H, (3.3.6)where H ∈ Ω ( M ) is the curvature of the gerbe.Let us conclude this section by mentioning the relation between this stack perspective on gen-eralised geometry and symplectic dg-geometry. Remark 3.26 (Relation with NQP-manifolds) . It is well-understood that, given a L ∞ -algebroid a ։ M , its Chevalley-Eilenberg dg-algebra CE( a ) can be seen as the dg-algebra of functions ona dg-manifold, also known as NQ-manifold. In the case of the Atiyah L ∞ -algebroid, we have CE( at G ) = (cid:16) C ∞ ( T ∗ [2] T [1] M ) , Q H (cid:17) (3.3.7)where the dg-manifold T ∗ [2] T [1] M , called Vinogradov algebroid, is canonically symplectic, i.e.it is a NQP-manifold. The Poisson bracket, combined with the differential Q H , reproduces theCourant -algebra [Roy02]. Inspired by this relation, a purely dg-geometric approach to DoubleField Theory was developed by [DS18, DHS18, CWIKW19, DS19]. In this subsection we will present an immediate application of the correspondence betweendoubled spaces and bundle gerbes. We will, indeed, formalise the NS5-brane of -dimensionalsupergravity as a topologically non-trivial higher Kaluza-Klein monopole on the bundle gerbe. Definition 3.27 (Higher Dirac monopole) . A higher Dirac monopole is a topologically non-trivial bundle gerbe G − ։ R , × (cid:0) R − { } (cid:1) .Here, R −{ } can be seen as the transverse space of the monopole and R , as its world-volume,magnetically charged by the Kalb-Ramond field. Remark 3.28 (Higher Dirac charge-quantization) . Notice that R − { } ≃ R + × S , where R + gives the radial direction in the transverse space and S the angular directions. Since R , × R + × S is homotopy equivalent to S , we have dd( G ) ∈ H ( S , Z ) ∼ = Z . This implies dd( G ) = m S )] , (3.4.1)with m ∈ Z , in direct analogy with the ordinary Dirac monopole.Now we can give a precise definition of a higher Kaluza-Klein monopole, which is constructedby directly generalising the ordinary Kaluza-Klein monopole [GP83] to a bundle gerbe. Definition 3.29 (Higher Kaluza-Klein monopole) . A higher Kaluza-Klein monopole [Alf20a] isa non-trivial bundle gerbe G − ։ R , × (cid:0) R − { } (cid:1) equipped with a generalised metric G such hat, on the atlas M , it takes the form G = η µν d x µ d x ν + η µν d e x µ d e x ν + h ( r ) δ ij d y i d y j + δ ij h ( r ) (d e y i + B ik d y k )(d e y j + B jk d y k ) (3.4.2)where the curvature of the gerbe and the harmonic function are respectively H = ⋆ R d h, h ( r ) = 1 + mr (3.4.3)with m ∈ Z and r := δ ij y i y j radius in the four dimensional transverse space. Here, the atlas ( M , ω, J ) of the bundle gerbe, with fundamental -form ω = d e x µ ∧ d x µ + (d e y i + B ij d y j ) ∧ d y i and { x µ , e x µ } are coordinates on T ∗ R , and { y i , e y i } are local coordinates on M| R −{ } .Notice that this monopole is nothing but a globally-defined Berman-Rudolph monopole [BR15].As observed by [BKM16], the Berman-Rudolph monopole gives rise to the non-geometric branes.In the global geometric context, the arising of non-geometric branes was studied in [Alf20a]. Remark 3.30 (NS5-brane is higher Kaluza-Klein monopole) . By higher Kaluza-Klein reductionof (3.4.2) to M = R , × R + × S we get the following metric and gerbe connection g = η µν d x µ d x ν + h ( r ) δ ij d y i d y j , B = B ij d y i ∧ d y j (3.4.4)which satisfy the conditions (3.4.3) on the transverse space. These are exactly the metric andKalb-Ramond field of an NS5-brane with H -charge m ∈ Z in d spacetime M .Therefore, the higher Kaluza-Klein monopole encompasses a higher Dirac monopole from def-inition 3.27, just as the Kaluza-Klein monopole does with an ordinary Dirac monopole. TheKaluza-Klein brane appears when spacetime is a non-trivial circle bundle and, analogously, theNS5-brane appears when the bundle gerbe is non-trivial. Recall that we already described a linearised version of T-duality in remark 3.9, where we showedthat every couple of T-dual Lie -algebras share the same atlas. Remark 4.1 (T-duality on the doubled space) . If two different bundle gerbes G and e G areT-dual, they will have the same atlas M . In other words, we will have a correspondence M G e G e ΦΦ (4.0.1)As we will show in the next subsection, the lift of the T-duality to the atlas will be an isometry ( M , J, ω ) −→ ( M , e J, e ω ) of para-Hermitian manifolds, i.e. a smooth map which preserves thepara-Hermitian metric η ( − , − ) := ω ( J − , − ) of M . -duality bundle gerbe bundle gerbeatlas atlas"doubled space"Figure 2: the "doubled space" seen as the atlas of both a bundle gerbe and its T-dual. Lemma 4.2 (Topological T-duality on the doubled space) . Let G Π −−→→ M and e G e Π −−→→ f M betwo topological T-dual bundle gerbes. Then their atlases, respectively ( M , J, ω ) and ( M , e J, e ω ) ,are related by a para-Hermitian isometry, i.e. a change of polarisation as defined by [MS18]. Proof.
Let us start from the T-duality diagram of two topologically T-dual bundle gerbes. Theatlas will sit on top of the diagram as it follows: M G × M f M M × M e GG M × M f M e G M f MM e ΦΠ π e π e ΠΠ π e π e Π π e π (4.1.1)Let us consider the atlas ( M , J, ω ) of the bundle gerbe G ։ M . Let e i ∈ Ω ( M ) be theconnection of the T n -bundle M ։ M . As shown in [Alf20a, p. 46], we can expand the local -form potential of the bundle gerbe in the connection e i ∈ Ω ( M ) by B ( α ) = B (0) ij e i ∧ e j + B (1)( α ) µi d x µ ∧ e i + B (2)( α ) µν d x µ ∧ d x ν (4.1.2)where B (0) ij is a globally defined scalar moduli field on M and, therefore, we omitted the α -index.The corresponding fundamental -form on the atlas M will be ω = (cid:0)e e i + B (0) ij e j (cid:1) ∧ e i + e e µ ∧ e µ , (4.1.3)where we patch-wise defined the following global -forms on the atlas: e µ = d x µ e i = d θ i ( α ) + A i ( α ) µ d x µ e e µ = d e x ( α ) µ + B (2)( α ) µν d x ν e e i = d e θ ( α ) i + B (1)( α ) iµ d x µ (4.1.4)Let us explicitly construct the para-Hermitian metric η of the atlas. This will globally be η ( − , − ) := ω ( J − , − ) ⇒ η = e e i ⊙ e i + e e µ ⊙ e µ (4.1.5)Since b := B (0) ij e i ∧ e j ∈ Ω ( M ) is a global -form, the moduli field B (0) ij ∈ C ∞ ( M, so ( n )) can be interpreted as a global B-shift. Thus, there exists an isometry of our para-Hermitianmanifold [MS18, p. 15] given by ω ′ = e b ω = e e i ∧ e i + e e µ ∧ e µ , (4.1.6)By using this isometry, we forgot the moduli field and we retained only the topologically relevantcomponent of the connection. Now, let ( M , e J, e ω ) be the atlas of the bundle gerbe e G ։ f M .Since we started from a couple of T-dual geometric backgrounds G and e G , we already knowthat the potential -form of the latter is e B ( α ) = e B (0) ij e e i ∧ e e j + A i ( α ) µ d x µ ∧ e e i + B (2)( α ) µν d x µ ∧ d x ν (4.1.7)where e B (0) ij is a global moduli field (which can be explicitly obtained by using the Buscherrules) and A i ( α ) µ is the -form potential of the T n -bundle M ։ M . Therefore, the T-dualcorresponding fundamental -form will be e ω = (cid:0) e i + e B (0) ij e e j (cid:1) ∧ e e i + e e µ ∧ e µ (4.1.8)Similarly to the first bundle gerbe, e b := e B (0) ij e e i ∧ e e j is a global -form and, thus, the map e ω ′ = e e b e ω = e i ∧ e e i + e e µ ∧ e µ (4.1.9)is an isometry of the para-Hermitian metric. Now, let us call J ′ and e J ′ the para-complexstructures corresponding to ω ′ and e ω ′ . We need to find a morphism of para-Hermitian manifolds f : ( M , J ′ , ω ′ ) −→ ( M , e J ′ , e ω ′ ) such that e ω ′ = f ∗ ω ′ and check that it is an isometry. This is mmediately the map f : (cid:0) x ( α ) , e x ( α ) , θ ( α ) , e θ ( α ) (cid:1) (cid:0) x ( α ) , e x ( α ) , e θ ( α ) , θ ( α ) (cid:1) , which is given by theexchange of the torus coordinates θ and e θ on each chart and is clearly an isometry. Therefore,by composition, we obtained an isometry e b ◦ f ◦ e − e b : ( M , J, ω ) −→ ( M , e J, e ω ) . Remark 4.3 (Buscher rules) . In [Alf20a, p. 47], we also showed that the Buscher transforma-tions ( g (0) ij , B (0) ij ) ( e g (0) ij , e B (0) ij ) of the moduli field of the metric and the Kalb-Ramond fieldfollow directly from applying the isometry of the lemma to the generalised metric, i.e. e G = f ∗ G . We identified the isometries of our atlas ( M , J, ω ) with changes of polarisation, i.e. with changesof T-duality frame. However, in general, it is not be possible to identify the image ( M , e J, e ω ) ofan isometry with the atlas of another bundle gerbe. In general, we can also obtain an almostpara-complex structure e J which is not integrable. In this case, the background described by thetransformed atlas ( M , e J, e ω ) is, then, a non-geometric background.For example, consider a bundle gerbe G ։ M such that (1) its base M ։ M is a T n -bundlewith connection e i and (2) its -form potential satisfies the equation L k i B ( α ) = 0 for vectorfields k i dual to e i . In [Alf20a, Alf20b], it is shown that we have a T-duality of the form M G × M K G K non-geometricbackground M M
0Φ Π π Π e ππ (4.2.1)where the dotted arrows are not actual maps, but just qualitative relations. Here, the generalisedcorrespondence space K is not anymore a T n × e T n -bundle on the base manifold M , but, inaccordance with [BHM07], it has the following bundle structure: e T n K c ( K ) := [ π ∗ H ] ∈ H ( M, Z ) T n M c ( M ) ∈ H ( M , Z ) M e ππ (4.2.2)where c ( − ) is the 1st Chern class and π ∗ is the fiber integration of the T n -bundle M . For a iscussion of these cases we redirect to [Alf20a, Alf20b]. The correspondence between doubled spaces and bundle gerbes we explored in this paper shedsnew light on the global geometry underlying Double Field Theory. Moreover, it provides a highergeometric explanation for the appearance of the extra coordinates and for para-Hermitian ge-ometry. These results are particularly important for the investigation of the other extendedgeometries, i.e. the exceptional geometries underlying Exceptional Field Theories, whose glob-alisation is significantly more obscure. In particular, the higher geometric perspective willallow to find a generalisation of para-Hermitian geometry for Exceptional Field Theory. Evenif exceptional generalised geometry [PPW08, CSCW11, CSCW14] is well-understood, such ageneralisation is still completely unknown. A generalised para-Hermitian formalism wouldbe extremely fruitful, for example, in the current research in exceptional Drinfel’d geome-tries [Sak20b, MT20, BTZ20, MS20, Sak20a, MST21].In [CP18,CP19a,CP19b], extended geometry has been studied in algebraic terms, in the light ofrepresentation theory. The extended/higher correspondence will then provide a complementaryglobal geometric perspective to extended geometry, as well as new connections between highergeometry and representation theory.Moreover, since the non-perturbative quantisation of strings and branes can be achieved byhigher geometric quantisation [SS11, BSS17, BS17, FRS16], the close relation we establishedbetween extended and higher geometries will have a profound impact on the problem of quan-tisation. This issue, among other ones, was started to be studied in [AB21].The higher structure which encompasses the global geometry of the C -field of -dimensionalsupergravity can be seen as a bundle -gerbe twisted by a bundle -gerbe [FSS15], which givesrise to the following diagram: G M5 ∗ G M2 B U (1) ∗ M B U (1) // B U (1) , B U (1) , Π M5 f M5 Π M2 f M2 / f M2 (5.0.1)where the twisted cocycle f M2 / can be also generalised to a -cohomotopy cocycle M f M2 / −−−−→ S , here the -sphere can be given in terms of its minimal Sullivan dg-algebra by CE( S ) = R [ g , g ] / h d g = 0 , d g + g ∧ g = 0 i , (5.0.2)where g and g are respectively - and -degree generators. In the context of L ∞ -superalgebras,a notion of super exceptional space R , | ex has been defined by [FSS18a,FSS19,FSS20]. Noticethat, in its bosonic form, i.e. R , = R , ⊕ ∧ ( R , ) ∗ ⊕ ∧ ( R , ) ∗ , (5.0.3)can also be interpreted as the atlas of the linearised version of the bundle gerbe G M5 ։ M . Ifwe split the base space in time and space by R , = R ⊕ R , we obtain the decomposition R , = R |{z} pp-wave ⊕ ∧ ( R ) ∗ | {z } M2-brane ⊕ ∧ R | {z } M9-brane ⊕ ∧ ( R ) ∗ | {z } M5-brane ⊕ ∧ R | {z } KK-monopole , (5.0.4)which agrees with the description of brane charges in M-theory [Hul98]. Notice that, if we splitthe base space in an internal and external space by R , = R , ⊕ R , we obtain R , = R , ⊕ (cid:16) R ⊕ ∧ ( R ) ∗ ⊕ ∧ ( R ) ∗ ⊕ ∧ R (cid:17) ⊕ · · · , (5.0.5)where the terms we explicitly wrote correspond to the (4 + 56) -dimensional extended spaceunderlying E Exceptional Field Theory [HS14]. Moreover, the terms we omitted are mixedterms involving wedge products between R , and R which correspond to tensor hierarchies[CCM19, HS19] at -degree. Moreover, as already argued by [AB18, sec. 9.2], the naturallyexpected structure generalising the fundamental -form to the exceptional case would generallybe an almost n -plectic structure. Recently, [SU20] proposed a local generalisation of the Born σ -model of the string to the M-branes. These are equipped with - and -forms which appearto be closely related to the transgression of the higher field whose curvature comes from the dg-algebra (5.0.2). All these are strong hints that the correspondence between extended geometryand higher geometry via atlases can be well-defined for the exceptional cases too. Acknowledgement
I want to thank the organisers Vicente Cortés, Liana David and Carlos Shahbazi of the work-shop
Generalised Geometry and Applications 2020 at Universität Hamburg. I would also liketo thank Christian Sämann, Franco Pezzella, Emanuel Malek, Urs Schreiber, Richard Szaboand Francesco Genovese for fruitful discussion. Finally, I want to thank Chris Blair and YuhoSakatani for interesting comments on the preprint of this paper.
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