Open exceptional strings and D-branes

We study D-branes in the extended geometry appearing in exceptional field theory (or exceptional generalised geometry). Starting from the exceptional sigma model (an $E_{d(d)}$ covariant worldsheet action with extra target space coordinates), we define open string boundary conditions. We write down Neumann and Dirichlet projectors compatible with the preservation of half-maximal supersymmetry by the brane (building on previous work on the definition of generalised orientifold quotients in exceptional field theory). This leads to a definition of D-branes, plus their S-duals, as particular subspaces of the exceptional geometry, and provides an opportunity to study D-branes in U-fold backgrounds.


Introduction
T-duality relates Dp-branes to D(p ± 1) branes, interchanging Neumann and Dirichlet boundary conditions on the string worldsheet. If one uses the doubled approach to the string worldsheet [1][2][3][4][5], an elegant picture emerges whereby all Dp-branes can be viewed as a single D-dimensional brane in the 2D-dimensional doubled target space: this can then intersect with the D-dimensional physical subspace in different number of directions in order to reproduce all standard p-branes [4,6,7]. In generalised geometry, [8,9], which underlies reformulations of supergravity such as [10] and the related formalism of double field theory (DFT) where the spacetime coordinates are doubled [11][12][13], this translates into the statement that D-branes are maximally isotropic subspaces of the doubled tangent bundle [9,14]. The purpose of this paper is to study the corresponding notion of D-branes in the exceptional geometry that appears in exceptional generalised geometry [15][16][17] and exceptional field theory (ExFT) [18][19][20][21][22][23][24][25][26]. We will combine insights from the exceptional sigma model [27,28] and from the realisation of orientifold quotients in exceptional field theory [29]. As O-planes and D-branes appear in type II theories alongside each other, we will use the realisation of the former as fixed points under reflections by Z 2 ⊂ E d(d) to write down projectors onto Dirichlet and Neumann directions in exceptional geometry. To ensure that we are describing D-branes, we will require compatibility with a string charge or string structure that appears in the exceptional sigma model. The crucial underlying feature common to both the orientifold and Dbrane projections is compatibility with an E d(d) half-maximal structure [30]: thus we can think of D-branes as defining what we might call half-maximal subspaces of the exceptional geometry.

Extended sigma models
We will begin with a string worldsheet action which corresponds to the doubled sigma model of [4,5] (see also [31]) and also to the exceptional sigma model of [27,28]. Some notation: σ A = (σ 0 , σ 1 ) are worldsheet coordinates. Target space coordinates, which are worldsheet scalars, come in two varieties: "external" X µ , µ = 1, . . . , n, and "extended" Y M , with the latter sitting in a representation, denoted by R 1 , of either O(D, D) or E d(d) .
Alongside these, we have also an auxiliary worldsheet one-form V M A . The worldsheet inverse metric is γ AB and the worldsheet alternating symbol is ǫ AB with ǫ 01 = −1. Then we write the action as We have the string charge q M N ∈R 2 which appears contracting the multiplet of two-forms in the Wess-Zumino term in (1). Finally,we have written These background fields can depend in principle on any of the extended coordinates Y M subject to a choice of solution of the section condition, which requires a limited coordinate dependence. This condition can be written as i.e. any combination of two derivatives acting on fields or products of fields must vanish when projected into theR 2 representation. It is common to introduce an invariant tensor Y M N KL proportional (for low enough d) to the projector onto the R 2 representation, such that the section condition is often written as Y M N P Q ∂ M ⊗ ∂ N = 0. This so-called Y-tensor appears in the definition of the generalised Lie derivative, [32], which defines the local symmetries of the background spacetime, namely E d (d) or O(D, D) valued diffeomorphisms associated to the coordinates Y M (rather than conventional GL(dim R 1 ) diffeomorphisms). A solution of the section condition is a choice of physical coordinates Y i ⊂ Y M on which the fields can depend such that (2) holds. In exceptional geometries, there is a d-dimensional solu-tion (corresponding to 11-dimensional M-theory), and inequivalent (d − 1)-dimensional solutions corresponding to the 10-dimensional IIA and IIB theories [19,33], while in doubled geometry the solutions are D-dimensional and again correspond to IIA or IIB.
The string charge q ∈R 2 appearing in the Wess-Zumino coupling of the action (1) is required in order to write down a coupling to the multiplet of two-forms B µν ∈ R 2 .
This charge obeys a constraint which should be thought as being solved after solving the section condition for ∂ M , and which guarantees gauge invariance of the action. This charge also appears in the "tension", which is given by with D = d − 1 for the E d(d) string. We will henceforth abbreviate T ≡ T (M, q). (1) is the auxiliary worldsheet one-form V M A , which is constrained such that V M A ∂ M = 0, again to be thought as being imposed after first solving the section condition. Integrating out the surviving components of V M A after solving this constraint eliminates the dual coordinates from the action, imposing a twisted duality constraint relating them to d − 1 physical coordinates, and reducing the action to the usual action for a string or 1-brane.

The final ingredient in
It is important to emphasise that the whole action, including the appearance of the auxiliary worldsheet one-form V M A and the charge constraint (3), follows from gauge invariance, assuming the natural coupling to the two-form B µν via q. For instance, invariance under the gauge transformation δB µν = ∂ ⊗Θ µν | R 2 , where Θ µν ∈ R 3 , inevitably requires (3).
Let us specify the precise details needed to specify the action (1) in the more familiar doubled case, and as the exceptional sigma model.
• Doubled string. We have Y M in the vector representation of O(D, D), so that R 1 = 2D. The section condition involves a projection onto R 2 = 1, and is equivalent to Writing Y M = (Y i ,Ỹ i ), the standard solution to the section condition is that We have Y M N P Q = η M N η P Q , and the charge can always be written as q M N = T F 1 η M N . As the generalised metric obeys M M N η N P M P Q = η M Q , the tension (4) reduces to T = T F 1 .
• Exceptional string: SL(5). This is the case when n = 7 and d = 4. The extended coordinates are in the antisymmetric representation of SL(5), thus we write them as Y ab = −Y ba , with a, b = 1, . . . , 5. We have R 2 =5, and the section condition is ef gh . The string charge is q a and obeys One three-dimensional solution of the section condition involves breaking SL(5) to GL(3). Letting a = (i, 4, 5) with i = 1, 2, 3, we take ∂ ab = (∂ i5 , 0). Then the only allowed charge is q i = q 5 = 0, q 4 = T F 1 . This describes a type IIA string with target space coordinates (X µ , Y i5 ), after integrating out the non-zero components of the auxiliary one-form. The tension is T = T F 1 .
The tension is proportional to T ∼ H αβ q α q β , where H αβ encodes the background dilaton and RR 0-form as an SL(2)/SO(2) coset element.
• Exceptional string: general results. In general, the extended coordinates decompose in terms of IIA and IIB physical and dual coordinates as follows: In the IIA case, the dual coordinates written here are conjugate to winding modes of the F1 string and Dp branes with p even. In the IIB case, they are conjugate to winding modes of the F1 string and Dp branes with p odd: in fact the F1 and D1 winding coordinates appear together as the SL(2) doubletỸ i α . There will also be coordinates conjugate to winding modes of the NS5 brane, Kaluza-Klein monopole, and (for high enough d) other "exotic" branes, denoted by the ellipsis in (7). The non-zero components of the charge, assuming the standard 10-dimensional solutions of the section condition, are always: (the S-duality SL(2) indices α, β can be raised and lowered using ǫ αβ ). Hence we always obtain the F1 action in IIA, and the (p, q) string action in IIB. Note that there are no solutions to the charge constraint (3) in the 11-dimensional solutions of the section constraint, as there are no strings in M-theory.

Boundary conditions
The realisation of doubled D-branes using the doubled sigma model was discussed in Hull's paper [4], and further studied in [6,7]. We now follow this approach and apply it to the exceptional sigma model (1). For simplicity, we restrict to backgrounds with We will in fact consider the sigma model in terms of the action supplemented by the constraint: This formulation is equivalent to that where the constraint is implemented by gauging the shift symmetry in dual directions (a consequence of the section condition), leading to the introduction of V M A as the gauge field for this symmetry [4,5,28,31]. (Note that one could view this, when the background metrics are flat, as describing the exceptional sigma model on the background R 1,n−1 ×T dim R 1 , i.e. on an "exceptional torus".) Varying (9) gives the following boundary terms: Let us now work in conformal gauge, γ 00 = −1 = −γ 11 , γ 01 = 0, ǫ 01 = −1. Our interest is in the boundary conditions for the extended coordinates Y M . For the time being we will assume Neumann boundary conditions for the X µ , that is ∂ 1 X µ = 0 at the worldsheet boundaries at σ = 0, π, and comment on the imposition of Dirichlet boundary conditions in these directions later (in section 2.3). So we are studying the boundary condition: At σ 1 = 0, let (P Dir ) M N denote the projector onto Dirichlet directions, and let (P Neu ) N M = δ M N − (P t Dir ) N M denote the Neumann projector. We have P Dir δY = 0 = δY P t Dir . This implies that we have to require at σ 1 = 0. (So note the Neumann projector naturally acts on M M N ∂ 1 Y N , hence its index structure). Compatibility with the constraint then means at σ 1 = 0. This can be achieved if which in turn implies P t Dir qP Dir = 0 = P Neu qP t Neu .
Evidently, at the other endpoint, σ 1 = π, we introduce similarly projectors P Dir and P Neu , which need not coincide with the ones at σ 1 = 0. Thus each endpoint of the string can be attached to a different subspace of the full extended space. (However, for the rest of this paper, we will assume that both endpoints of the string obey identical boundary conditions.) When we are dealing with the doubled string, the situation is geometrically appealing.
Note for O(D, D), we have q M N = η M N (setting the tension T F 1 to 1), which is invertible, so that so Dirichlet and Neumann projectors are mapped into each other by applying η. Equivalently, for every Dirichlet direction we have a Neumann direction, reflecting the fact that T-duality interchanges these boundary conditions. A doubled D-brane then amounts to a D-dimensional subspace of the 2D-dimensional doubled space, and the canonical form of the projectors P Dir and P Neu is Depending on how one chooses the physical coordinates, the doubled D-brane will intersect with the D-dimensional physical subspace in differing numbers of directions, and so realises the full set of expected p-branes.
For the exceptional sigma model, the string charge q M N will not be invertible. The Note that (when either q = 0 or p = 0 in the IIB case) this is an embedding of the SO(D, D) structure into E d(d) language (breaking the latter to the former). In order to find Neumann and Dirichlet projectors obeying (16) for q M N of the form (20), we will use some additional information.

O, D
String theory also contains orientifold planes, which are (non-dynamical) extended objects carrying (negative) RR charge, and which appear alongside D-branes of the same dimensionality (as required for charge cancellation). In particular IIA contains Op planes and Dp branes with p even, while in IIB we have p odd (we only consider stable p-branes).
An elegant description of orientifold quotients (at the supergravity level) in exceptional field theory was developed in [29]. For the standard orientifolds, we consider a quo- , with this Z 2 acting "geometrically" on the fields and coordinates of exceptional field theory according to how they transform as representations of E d(d) .
The fixed points of this Z 2 define generalised orientifold planes as subspaces of the ex- The connection can be formalised using supersymmetry. Exceptional field theory describes maximal supergravity in 11 dimensions and lower (see for example [34]). In order to describe backgrounds which break some supersymmetry, or truncations to theories with less supersymmetry, an E d(d) covariant notion of a half-maximal structure can be defined [30]. This is a set of generalised tensors, globally defined on the physical spacetime underlying the exceptional field theory construction, obeying certain compatibility conditions, whose existence is equivalent to that of a set of Killing spinors implying the presence of half-maximal supersymmetry.
The generalised orientifold quotients (or "O-folds") considered in [29] are restricted by the requirement that they preserve the existence of the E d(d) half-maximal structure, and thus lead to configurations with half the supersymmetry. The important point for us now is that D-branes themselves are of course half-BPS objects; this underlies how they can appear alongside O-planes in half-maximal theories (type I and its T-duals).
Putting O and D together, we propose that we can use the Z 2 transformation of [29] to define the correct Dirichlet and Neumann projectors obeying (16), and which describe therefore D-branes as "half-maximal subspaces" in the exceptional geometry of ExFT.

D-brane structure
Now we give a formal definition of what we might choose to call a D-brane structure in exceptional geometry. By exceptional geometry we mean either that appearing in exceptional field theory [18,19] or alternatively in exceptional generalised geometry [15][16][17]. In exceptional field theory, we have fields depending on the extended coordinates (X µ , Y M ), and generalised vectors and tensors transforming in the representations, In exceptional generalised geometry, we work with a generalised tangent bundle E over a base manifold M , and this generalised tangent bundle carries an action of . To be precise, here we would take M to be a (d − 1)-dimensional manifold and define a set of bundles R 1 , R 2 , R 3 , . . . , such that generalised tensors transforming in the with the even/odd antisymmetric products corresponding to IIA and IIB respectively, where the ellipsis denotes additional factors needed for d = 7 [35]. We mentioned earlier that the representation R 2 is contained within the symmetric tensor product of R 1 with itself. This fact allows us to define a (symmetric) product ∧ : takes a pair of sections of R 1 and projects them into a section of R 2 , which we will use below.
We can think of the extended geometry of ExFT as being locally isomorphic to the extended tangent bundle E. We will therefore describe our D-brane structure in terms of maps on E. In both cases, we will write partial derivatives ∂ M . In ExFT, we think of the choice of solution of the section condition as telling us which components of these are non-zero, corresponding to derivatives with respect to the physical coordinates.
Then different choices of this solution correspond to IIA versus IIB. In exceptional generalised geometry, we think of the physical coordinates of the underlying manifold M as being embedded into ∂ M with all other components zero. Then different choices of this embedding are used for IIA versus IIB. The data we use to specify a D-brane structure in exceptional geometry consists of: • an involution Z : E → E, Z 2 = 1, which defines projectors • a section of theR 2 bundle, q, obeying the string charge condition We can use this to define a degenerate bilinear form (which we could also see as a non-invertible map from the exceptional tangent bundle E to its dual E * , q : E → E * ) using the symmetric map ∧ : where on the right hand side we use the natural pairing between sections of R 2 andR 2 , denoting this by a dot. We require that (this is the condition (16) arising from the worldsheet boundary conditions and self-duality constraint) for arbitrary U, V ∈ Γ(E), or equivalently that This implies that (this is (17)) i.e. both the images of the projectors are null with respect to the string charge q.
• an E d(d) half-maximal structure [30], consisting of d − 1 generalised vectors J u ∈ Γ(E), u = 1, . . . , d − 1, and generalised tensors, K ∈ Γ(R 2 ),K ∈ Γ(R 2 ), obeying certain compatibility conditions, and such that they are contained in the image of the Neumann projector, i.e. they are invariant under the involution. Note the the compatibility conditions include K ·K > 0 and J u ∧ J v = δ uv K. As we have firstly that q(J u , J v ) = 0, it also follows that q · (J u ∧ J v ) = δ uv q · K = 0. Roughly speaking, both q and (K,K) define separate SO(D, D) structures whose intersection determines the orientation of the D-brane in the physical subspace. We have some further comments on this in appendix B.
As in [6,7], we can also require the Dirichlet and Neumann projectors to be orthogonal with respect to the generalised metric, and that the Neumann subbundle is integrable, where L is the generalised Lie derivative. In components, given the transformation Z M N squaring to the identity, the crucial conditions (24) and (27), become and the preservation of the half-maximal structure J u M , K M N ,K M N is that In practice, we can find Z M N as in [29] by picking a suitable form for the half-maximal structure and working out the action of its stabiliser subgroup within E d(d) . Then we specialise to a Z 2 discrete subgroup of this stabiliser. This was worked out explicitly for SL (5) and SO(5, 5) in [29] but applies for higher rank groups too (note that the details of the half-maximal structure are slightly different in d = 7 [30]). In the next subsection, we will use the results on SL(5) to explore how the above definitions work out in an explicit example.
First, we can already give some general expressions. In particular, we can write down sentations by using the results of appendix B of [29]. An explicit realisation of this (as in [36]) involves introducing creation and anni- Defining a vacuum |0 such that ψ i |0 = 0, we build spinors of definite chirality by acting with an even or odd number of ψ i on |0 . Then when the Z 2 transformations acts on the doubled vector representation 2D as Z = diag(I p , −I n , −I p , I n ) (corresponding to n Dirichlet directions and p Neumann directions in the D-dimensional physical space), it acts on the spinor as the operator 1 where we split i = (µ, a) corresponding to the even and odd physical directions, and N (p) and N (n) denote the number operators for the (ψ µ , ψ µ ) and (ψ a , ψ a ) spinor subspaces.
On a spinor state we haveẐ Note that the action of Z andẐ does not correspond to a transformation in O(D, D) or its double cover. In particular, Z sends the O(D, D) structure η M N to −η M N . Despite this, they represent a symmetry of the doubled geometry, preserving the action and local symmetry transformations (in which η M N appears alongside its inverse: the combination of the pair is invariant under the action of Z). Though Z always squares to one, we havê N F is even/odd and n is odd/even. This picks out the conventional spinor and p-form chiralities along with the correct D-brane dimensions in IIA and IIB.
By taking n = 0, so that Z = diag(I D , −I D ) on the 2D representation, we obtain where N F is the total number operator. This must be odd, so this is an action in IIB (in particular defining 9-branes). Then it is easy to see thatZ acts as +1 on the p-forms with p = 1, 5, 9 and as −1 on the p forms with p = 3, 7. It follows that acting on 2D ⊕ 2 D−1 there are thus D + D 3 + D 7 Dirichlet directions.
56 6 20 6 32 valued. This reveals that for E 6(6) , the Z 2 must end up acting as −1 on the representation r = 1, and for E 7(7) it must act as it does on the 2D representation on r = 2D. For E 8(8) , a large number of additional dual fields make an appearance, so this method does not immediately tell us the answer. We can note however the general pattern is that there are always 2 D−1 Dirichlet directions, as shown in table 1.
Let's take stock of the general situation before we move on to an explicit example in SL(5).
We have a Z 2 transformation acting on the exceptional geometry of ExFT or generalised geometry, corresponding to that used originally in [29] to define a generalised orientifold quotient. This Z 2 preserves half-maximal supersymmetry. We can use it to define a pair of projectors, P Dir = 1 2 (1 − Z) and P t Neu = 1 2 (1 + Z), which can be used to define Dirichlet and Neumann boundary conditions for the exceptional sigma model. Finally, we should note that the projectors P Dir and P Neu can be used to define these half-maximal subspaces wrapped by branes also when the compatibility condition with q is not obeyed. In this case, the branes would not be interpreted as D-branes in spacetime.
Rather, they may be viewed as some sort of NSNS brane related to the existence of the heterotic theories, as suggested in [37], or indeed as the Hořava-Witten end-of-the-world branes [38,39], as implied by the generalised orientifold analysis of [29].

Example: SL(5)
The d = 4 SL(5) exceptional geometry [18,25,33,40] is an instructive example, and one in which it is simple to enumerate all possibilities. We denote five-dimensional fundamental indices by a, b. The Z 2 involution obtained in [29] (and which we know from the discussion there is compatible with the existence of the half-maximal structure) can be taken to be: where i, j = 1, . . . , 4 label the four odd components of generalised tensors transforming in the 5 or5. Thus we write a = (i, s) with Z s s = +1. The R 1 representation is the antisymmetric, thus generalised vectors V M are written as V ab with V ab = −V ba , and we have Z M N given by and projectors These can therefore be written in the canonical form (Note that our contraction convention is This means that the components V ij of a generalised vector V M are even under Z, while the components V is are odd. The string charge or string structure is q a or q ab,cd = ǫ abcde q e . The condition (31) imposes that q s = 0 or equivalently q ij,kl = 0. The condition (30) requires that the physical coordinates are embedded into ∂ ab such that We analyse what D-branes are possible by looking in turn at IIA and IIB embeddings, and seeing what are the consequences for writing down Z a b as in (36)  this is maybe more manifest in the former picture (where we fix the brane definition and rotate the section). However, we will adopt the language of the latter in practice, fixing our notation for the physical solution of the section condition and changing the order of the indices in Z a b . In terms of the extended geometry, we think of the branes as fixed in the Y is directions and therefore spanning the (X µ , Y ij ) directions.
• IIA: let a = (i, 4, 5) with s = 5, i = (i, 4). In this and all IIA cases, the nonvanishing components of ∂ M are ∂ i5 . The string charge q i = (0, q 4 , 0) obeys the defining conditions. Therefore we obtain a D-brane, with the Dirichlet projector acting on spacetime vectors V i5 as −I. We may also note that the coordinates Y ij are all dual coordinates. Thus the D-brane is extended in the directions X µ alone: this is therefore a D6 brane.
• IIA: let a = (i, 4, 5) with s = 4, i = (i, 5). The string charge is forced to vanish: this is not a valid definition of a D-brane in type II theories. Referring to the classification in [29], we see that in fact the Z transformation here would lead to the heterotic E 8 × E 8 theory when applied as a quotient. There are no D-branes in the heterotic theories, so this is consistent. Another way to say the same thing is to note that the physical directions (X µ , Y i5 ) are all even and therefore the brane in this case would be spacetime filling, but IIA does not have D9 branes. One can also see that the M-theory direction Y 45 is odd under the Z 2 -in the orientifold picture, this is the Hořava-Witten interval.

Dirichlet conditions in external directions
We assumed in section 1.2 that the external directions X µ obeyed Neumann boundary conditions, so that the branes we are considering span the entire external space of the full exceptional geometry (X µ , Y M ). Naively, it might seem that one could instead have the X µ obey a mix of Neumann and Dirichlet conditions, resulting in p-branes for arbitrary p in both the IIA and IIB embeddings! To rule out the wrong p branes in each case, it is likely that one needs to return to the condition of half-maximal supersymmetry anew, and check what happens when acting with an additional Z 2 reflection on an external direction. We will not attempt this analysis but rather offer one proposal to obtain the correct branes.
We suppose we pick a single X µ direction to be Dirichlet. coordinates and view the exchange of these coordinates an an outer automorphism as in [42]. For example, for SL(5), we decompose Y ab = (Y i5 , Y ij , Y i4 , Y 45 ) and the outer automorphism acts by swapping Y i5 ↔ Y ij . Let us call this transformation σ. We propose that switching a single X µ from Neumann to Dirichlet corresponds to acting on the exceptional geometry with the transformation σ, Y → σ(Y ). Thus, in particular, Then we have Clearly if we then impose Dirichlet boundary conditions on a second external coordinate, we apply σ again, but σ 2 = 1, so this brings us back to the original situation.
Hence the results of the previous subsection for Dp branes (but not the S-duals on the IIB side) in SL(5) can be interpreted as holding "modulo 2". In this way, we can indeed define D-branes with any number of external Dirichlet directions: when this number is odd, we have to use the additional symmetry σ.

D-branes in S-and U-folds
We will now apply our definition of D-branes in the SL(5) ExFT to the situation where we have some non-trivial U-duality monodromy, and want to know which D-branes are compatible with this monodromy. This is a step towards understanding D-branes in U-folds, and generalises the T-fold analysis of [6,7]. As a proof of concept, we will focus 4 In principle, suppose we start with the D7 and consider its S-dual 7-brane, on which D1 branes end. T-dualising this along worldvolume directions will lead to p-branes, p < 7, which are not D-branes (they will depend on the string coupling gs as g α s for α < −1), and on which Dp ′ branes, p ′ > 1, end. These will not be found when our string charge qMN obeys the constraint (30) and corresponds to 1-branes in 10-dimensions. However, on assuming isometries we may be able to describe more general branes ending on branes -see the comment in the final discussion -which should follow from relaxing the restriction here to q invariant under σ.
on some illustrative examples in the IIB case, and leave an exhaustive classification for future work.

SL(5) duality in IIB
We will focus on the 10-dimensional extended geometry of the SL(5) ExFT, described by the generalised metric M M N , which here can be written as M ab,cd = 2m a[c m d]b in terms of a symmetric unit determinant "little metric" m ab [40,43]. In the IIB solution of the section condition [33], we parametrise a generalised tensor V a as V a = (V i , V α ), with i = 1, 2, 3, and α = 4, 5. This unusual convention for the GL(3) index i is such Here g ij is the inverse spacetime metric,ṽ iα ≡ 1 2 η ijkC jk α corresponds to the RR/NSNS 2-form doublet, with C ij α = (C ij , B ij ), and contains the dilaton and RR 0-form.

U-duality transformations act such that
Geometric U-dualities include shifts of the two-forms,ṽ iα →ṽ iα + Ω iα , and GL (3) coordinate transformations generated by We also have non-geometric U-dualities which shift "bivectors", generated by and SL(2) S-dualities

SL(5) U-folds in IIB
We want to consider configurations where the exceptional geometry is patched by SL (5) transformations. The stereotypical situation is that our fields depend on some periodic coordinate θ, and we have a monodromy m ab (θ + 2π) = (U t ) a c m cd (θ)U d b . For instance, we could tread the well-worn path of considering a three-torus with flux of the NSNS two-form (as inspired by [44,45] and here essentially following the duality chains in [46]): We have written the 10-dimensional Einstein frame metricĝμν in a 7 + 3 split of the coordinates,μ = (µ, i), with "external" directions µ = 0, . . . , 6 and "internal" directions i = 1, 2, 3, which is appropriate for the SL(5) ExFT. Our D-brane conditions will give us information about branes wrapping the internal directions.
The metric components appearing in the IIB generalised metric are g ij ≡ĝ ij . In the absence of off-diagonal components, the combination g µν ≡ (det g ij ) 1/5ĝ µν is invariant under SL(5) U-duality transformations.
The little metric for the background (48) is: We will generate a new background by U-dualising with the transformation which in fact amounts to T-dualising in the y 1 , y 2 directions. This leads to the Einstein frame configuration (the quantity in square brackets is the string frame metric): 5 This is non-geometric for y 3 → y 3 + 2π, and transforms as a T-fold under an SO(2, 2) duality transformation embedded in SL(5) as a U ω of the form (46) with Acting with S-duality (47) with b = 1, c = −1, a = d = 0, on the configuration (51) trivially generates a genuine U-fold, ds 2 E = e Φ/2 η µν dx µ dx ν + (dy 3 ) 2 + 1 1 + (Hy 3 ) 2 ((dy 1 ) 2 + (dy 2 ) 2 ) , with the U-fold monodromy again of the form (46) with Alternatively, we could S-dualise the original configuration (48), and then act with (50), leading to a configuration with a flat background metric, vanishing two-forms, and H αβ = 1 Hy 3 This is an S-fold in that the monodromy lies in the SL(2) S-duality subgroup, and amounts to shifting C 0 → C 0 + 2πH. 6 If we act with the fundamental S-duality again, we get which is a "non-geometric" S-fold with a = d = 1, b = 0 and c = −2πH.
All the above configurations are meant to be illustrative examples of these standard monodromies. One can generate more realistic backgrounds by starting with the solution for the NS5 brane in place of the three-torus with H-flux given in (48), smearing twice in transverse directions and then dualising as above. Such chains of dualities are also discussed in [46], and lead to non-geometric exotic branes [47].

Monodromies, string charges and D-branes
Now let's discuss how to make statements about strings and D-branes in U-folds using the previous discussion. In SL(5) exceptional geometry, we needed to combine the string charge q a with the Z 2 transformation Z a b in order to define D-branes. We suppose that we can apply this definition to backgrounds with non-trivial SL(5) monodromies. Under We want to see how this affects the Dirichlet boundary conditions defined using the Z 2 transformation. For consistency, we require that the monodromy lead to the same boundary conditions. 7 In the IIB case, the non-zero components of the string charge lay in the SL(2) directions, thus q a = (0, q α ). This is trivially preserved (up to a scaling in the case of U A ) under geometric U-dualities (45). Let us therefore consider the more interesting situations where we have monodromies lying in the S-duality subgroup or of the nongeometric type (46).
Under S-dualities, we have q α → S α β q β . The Z 2 leading to D7 branes wrapping a single direction of the internal space is Z a b = diag(−1, −1, +1, −1, −1). This is clearly preserved by S-duality transformations (47). Altogether the pair (q α , Z a b ) transform under an S-duality monodromy to (S α β q β , Z a b ). One can consider this as telling us that the 7-brane on which (p, q) strings with charge q α end is transformed into the 7-brane on which the (p, q) strings with charge S α β q β end. This is what one would expect.
The Z 2 leading to 9-branes wrapping all three directions of the internal space is . The S-duality transformation (47) turns this into This leads to a Dirichlet projector such that (we label a = (i, 4, 5) and do not lower the i = 1, 2, 3 indices for convenience) (P Dir δY ) ij = 0 , With the upper sign, the original Dirichlet condition was δY i4 = 0 = δY 45  space in the non-geometric S-fold (56) for which cd = 0. However, they would be allowed in the former case (upper sign), as b = 0. Additonally, the monodromy will lead to a mixed type of string/brane combination (that is, some mixture of F1/D9 and D1/NS9) unless q α is preserved.
Next, we consider the non-geometric U-dualities (46). Under these, we have q a → (ω iβ q β , q α ). The transformed charge q a will in this case not obey the charge condition Dualising the indices on ω iα , this is the same as q α ω ij α ∂ k = 0. In this dualised form, ω ij α can be seen as the shift in a bivectorC ij α . For generic q α , this tells us that in order to have well-defined strings we need the indices i, j for which the bivector has non-zero components to correspond to isometry directions.
(This condition is frequently used for the bivector in the NSNS sector [48].) Observe that this is the case in the backgrounds (51) and (53), for which we have ω 12 α = 0, after T-dualising on the y 1 and y 2 directions, which were isometries.
We now consider the action of U ω on the transformation Z a b . In general, we find that the monodromy turns Z a b into Clearly, Z a b will be preserved if the top-right block vanishes, so that the boundary conditions are trivially invariant under the monodromy. For the D7 case, we have Z α β = −δ α β , and Z i j acting as −1 in two directions and +1 in the other direction. Then the the top-right block of (59) is non-zero if ω i + α = 0, where i + denotes this even direction.
For the D9 case, we have Z i j = −δ i j and Z α β = diag(±1, ∓1), so this is non-vanishing if either ω i4 or ω i5 is non-vanishing, depending on the sign choice.
Now examine the D9 case, for where we take ω iα to have only one non-vanishing component in order to preserve the string charge (assuming isometries as above). We yet again analyse the Dirichlet projection condition, finding that (P Dir δY ) 12 = (P Dir δY ) 14 = (P Dir δY ) 24 = 0 , These are consistent with the original projection conditions: therefore we can have threebranes in the T-fold background (51), again agreeing with [6]. S-duality interchanges the 4 and 5 indices, and shows that these three-branes are also possible in the U-fold S-dual to (51).
It is clear how to continue this analysis for other monodromies, and also in the IIA case. We hope that the above discussion demonstrates the general situation adequately.

Discussion
In this short paper, we have scratched the surface of the topic of D-branes, and some of their S-duals, in exceptional geometry. This involved combining previous work on strings whose target space is this exceptional geometry [27,28] with the study of generalised orientifolds in ExFT [29], providing a promising route in to the study of D-branes in this setting. We would like to propose a number of developments one could now attempt building on this work.
More on D-branes. We only studied the simplest examples of D-branes in this paper.
One could say much more about their presence or absence in U-folds. For instance, we did not consider locally non-geometric examples, where the background spacetime depends explicitly on dual coordinates. The description of D-branes in T-folds was recently revisited in [49,50] in order to take decoupling limits leading to non-commutative and non-associative theories on the D-brane worldvolume: we should explore how the obvious generalisations to U-folds may work.
We briefly mentioned the possibility of having different boundary conditions at the string endpoints, attaching each end of the string to separate subspaces of the exceptional geometry. This would involve a pair of projectors, P Dir , P Dir , each compatible with the same string charge q but in general preserving different half-maximal structures. Overall this would generically give a configuration with less SUSY. There is then likely a neat classification of such intersecting brane configurations available with this approach.
It might also be possible to study D-branes in so-called non-Riemannian backgrounds [51,52], where the generalised metric cannot be parametrised in terms of an invertible spacetime metric: this might allow a novel way to define D-branes in non-relativistic theories, for instance.
Heterotic strings? In the IIB case, our definition led also to the S-duals of the usual D-branes. This included not only the (p, q) 7-branes, but also an S-dual of the D9 brane.
This would be an NS9 brane on which open D1 branes end. This should correspond in fact to the heterotic SO(32) string, and so it is natural to ask whether the open string version of the exceptional sigma model provides a novel and perhaps unexpected duality symmetric treatment of the type I and heterotic strings, combining insights from this paper with the results of [29,41].
Branes ending on branes. We had string charges q obeying the constraint q⊗∂| R 3 = 0, which we solved assuming the derivative ∂ M corresponded to the solutions of the section condition giving 10-dimensional IIA or IIB. In principle, if we assume isometries, so that ∂ M = 0, then the charge q is unconstrained, and describes the full E d(d) multiplet of strings obtained in (11 − d) dimensions by partially wrapping branes on a T d−1 torus. In this case, the definition for D-branes we used may also describe the embedding into the exceptional geometry of the more general set of branes on which these partially wrapped branes can end.
One could potentially also proceed to study higher rank branes directly. For instance, membranes in exceptional geometry must be characterised by a chargeq ∈R 3 , obeying constraints such asq ⊗ ∂| R 4 = 0. Requiring for exampleq M N P = −Z K M Z L N Z Q PqKLQ with the same Z 2 may then allow us to obtain exceptional geometric definitions of the branes on which membranes end. Indeed, for SL(5) this charge can be seen to case which may be applicable include [14,53,54] (see also [55] for (NS)5-branes in DFT and [56] in ExFT). In fact, the paper [57] has already described U-duality covariant expressions for the Wess-Zumino terms of D-branes in various dimensions. Interestingly, this involved a doubling of the number of worldvolume scalars corresponding to internal directions. In an approach based on the exceptional geometry of ExFT, we would want to embed these doubled coordinates into the full extended coordinates Y M (this may be reminiscent of how the exceptional sigma model contains a reduction to the doubled sigma model [28]), and also to understand the Y M -dependent gauge transformations of the generalised gauge field to which the brane will couple electrically. The most natural case to consider is that of branes which are external spacetime filling, and so couple to an E d(d) multiplet of forms C µ 1 ...µn which lies beyond the usual tensor hierarchy construction needed in ExFT. The E d(d) representations of these forms and the structure of the charges to which they couple (the generalisations of the string charge q appearing in the exceptional sigma model) have been described in [57].
More on the geometry. We would also like to obtain a more comprehensive understanding of the geometry of the subspaces defined by our projectors. The numerology of the number of Dirichlet directions is quite appealing (see table 1) in this regard. There may also be more to say about the interplay between the string charge q and the halfmaximal structure (see appendix B). Another observation is the following. In doubled geometry, one can view the D-dimensional D-brane as well as the physical subspace as maximally isotropic subspaces of the 2D-dimensional space. This way of viewing the physical subspace is important for generalised dualities using the notion of a Drinfeld double [58]. Perhaps similar structures, and generalised generalised dualities, are implied by the branes in exceptional geometry.

B Comments on half-maximal structures and O(D, D)
Our definition of the D-brane structure in exceptional geometry in section 2.1 included compatibility with a Neumann projected half-maximal structure, involving J u ∈ Γ(R 1 ) and K ∈ Γ(R 2 ), such that q(J u , J v ) = 0, q · K = 0. In [30], It is also interesting to consider the explicit reduction to O(D, D) of our definition.
Consider again SL(5), let the 5-dimensional fundamental index a = (i, 4, 5) and fix q 4 = 0, q i = q 5 = 0, so that the string charge defines F1 strings (in both IIA and IIB). Because we require Z a b q b = −q a but Z b a K b = K b , this means we have K 4 = 0.
Normally, the idea is to fix K ♯ = 0, K α = 0, for a = (α, ♯), such that the compatibility condition J u ∧ J v = δ uv K becomes Then picking J α♯ u = 0, and rescaling J αβ u by a power of K ♯ (which is proportional to the generalised dilaton e −2d ), this is equivalent to J M u J N v η M N = δ uv after splitting α = (i, 5) with J M u = (J i5 u , J ij u ), i, j = 1, 2, 3. This gives the O(D, D) half-maximal structure selected in [30]. For us, the "physical" coordinates are fixed by the choice of q a (via the string charge constraint (30) involving an off-diagonal block of the O(3, 3) gamma matrices. As J M u in (71) is meant to be Neumann projected, we can see that when the physical coordinates are Y ij , i.e.
in IIB, this corresponds to a D9 brane (because P Neu = diag(I, 0)), while when the physical coordinates are Y i5 , i.e. in IIA, this corresponds to a D6 brane (because P Neu = diag(0, I)). Taking K i = 0 gives again the conditions (72) with particular forms of J u and K corresponding to the D7 and D8 cases.