Born sigma model for branes in exceptional geometry

In double field theory, the physical space has been understood as a subspace of the doubled space. Recently, the doubled space is defined as the para-Hermitian manifold and the physical space is realized as a leaf of a foliation of the doubled space. This construction naturally introduces the fundamental 2-form, which plays an important role in a reformulation of string theory known as the Born sigma model. In this paper, we present the Born sigma model for $p$-branes in M-theory and type IIB theory by extending the fundamental 2-form into U-duality-covariant $(p+1)$-forms.


Introduction
Double Field Theory (DFT) [1][2][3][4][5] has been developed for the T -duality-covariant formulation of supergravity. This is defined on a 2d-dimensional space called the doubled space, but in order to consistently formulate the theory, a constraint known as the section condition is required. Accordingly, we require that any fields depend only on at most d coordinates.
Namely, we suppose that any fields are defined on a d-dimensional physical space. The choice of the physical directions is arbitrary and it can be specified by the polarization tensor [6].
This arbitrariness in the choice of the polarization tensor can be understood as T -duality symmetry. Although the polarization tensor is an important object in the doubled space, it appears only implicitly in the conventional formulation of DFT. This is because we usually consider a specific class of physical spaces. As we review in section 2, if we consider a general polarization, the polarization tensor appears explicitly, for example, in the definition of the generalized Lie derivative. This kind of general polarizations will be important to investigate non-trivial applications of DFT, and various aspects have been studied recently in [7][8][9][10][11][12][13].
In this recent approach, the physical space is specified by the polarization tensor, or equivalently, an almost para-complex structure K I J satisfying (K 2 ) I J = δ I J (see [14] for details of the para-complex structure). A set of d eigenvectors with the eigenvalue +1 spans the tangent space of the physical space while a set of d eigenvectors with the eigenvalue −1 spans the unphysical gauge orbits in the sense of [15]. The polarization tensors Π ± , which pick out the physical/unphysical directions, are defined as (Π ± ) I J ≡ 1 2 δ I J ±K I J and the section condition is expressed as (Π − ) J I ∂ J f (x) = 0 . When K I J satisfies the integrability condition, it is called a para-complex structure, and it allows us to find a local coordinate system (x I ) = (x m ,x m ) such that the physical and the unphysical space are realized asx m = const. and x m = const., respectively. Then, K I J takes the form and can be interpreted as a symplectic form. Accordingly, the interpretation of the doubled space as a kind of phase space has been developed in [16][17][18][19][20][21], and there, the T -duality is interpreted as the Fourier transformation, x m → p m and p m → −x m . The symmetry under the Fourier transformation is known as the Born reciprocity, and the doubled space equipped with a certain dynamical metric H IJ is called the Born manifold. Apart from DFT, the fundamental 2-form ω appears also in the string action, known as the Tseytlin action [22,23] The term including ω is topological and has not been included originally in [22,23], but its importance has been discussed in [17,[24][25][26][27]. The topological term is introduced also in Hull's approach [6,25] where the worldsheet covariance is manifest. More recently, a duality-covariant string action in an arbitrary curved background is provided in [28]. Subsequently, by further adding a certain total-derivative term, a duality-covariant action called the gauged Born sigma model has been proposed in [29], which explicitly utilizes ω.
In this paper, we study an extension of the Born sigma model for various p-branes in M-theory and type IIB theory. It terns out that this action is the same as the one studied in [30,31] but the reformulation using ω makes the structure simpler and clearer. In addition, the rank of the E n U -duality group has been assumed to satisfy n ≤ 7 in [30,31], but here it is not assumed and we can consider the full theory n = 11 . Moreover, regarding type IIB branes, only the (p, q)-string has been explicitly considered in [31] but here we also provide the D3-brane action and the action for the (p, q)-five brane.
This paper is organized as follows. In section 2, we review the description of the doubled space as the para-Hermitian manifold. After reviewing the geometric framework, in section 2.4, we explain our approach to string sigma model. In section 3, we apply a similar discussion to the case of the exceptional space, and study sigma model actions for various p-branes. In section 4, in order to clarify the relation to Tseytlin's approach, we study the brane actions in Hamiltonian form. Section 5 is devoted to the summary and discussion.
2 Para-Hermitial geometry for double field theory In this section, we briefly review the para-Hermitian geometry from a physics point of view.
We then explain our approach to brane actions by using the string sigma model as a prototype.

Para-Hermitial geometry
In DFT, we consider a doubled space which is a smooth 2d-dimensional manifold M endowed with a metric η IJ of signature (d, d). We may introduce the standard Christoffel symbol Γ I J K associated with η IJ and denote the covariant derivative as We assume that the metric η IJ is flat and always work with a local coordinate system (x I ) = (x m ,x m ) where η IJ and its inverse η IJ have the form 1 In order to consistently formulate DFT, we require that any two of the fields and gauge parameters, say f and g, satisfy the section condition Here, the covariant derivative∇ I can be replaced by ∂ I because we are assuming Eq. (2.2) where Γ I J K = 0. In the following, we review that the section condition suggests us to regard the doubled space as a para-Hermitian manifold [7][8][9][10][11][12][13].
The section condition (2.3) indicates that the derivatives of any fields lie in a common null subspace. Accordingly, we introduce a projection operator Π + of rank d satisfying [6] which is known as the polarization tensor. Then we assume that any fields f satisfẙ We can easily check that the section condition (2.3) is indeed satisfied under this situation: If we also introduce a projection operator Π − ≡ 1 − Π + onto the orthogonal directions, Eq. (2.5) can be also expressed as Then using the two projectors Π ± , we can construct an almost para-complex structure (or an almost product structure) By using the polarization tensors, we define two distributions L * ± , spanned at each point by (Π ± ) I J dx J . The condition for each of the distributions to be integrable is given by If we define for arbitrary vector fields V I and W I , the integrability conditions are equivalent to N ± = 0 .
If both of these distributions are integrable, we can find a local coordinate system (x m ,x m ) where Π ± and K have the form Such coordinates x m /x m are called para-holomorphic/para-anti-holomorphic coordinates, where L * + /L * − are respectively spanned by dx m /dx m . On an overlap of two such coordinate patches (U α , x I (α) ) and (U β , x I (β) ) with U α ∩ U β = ∅ , by requiring the para-Cauchy-Riemann equation we can consistently define the para-complex structure in both patches as (2.11) and K can be globally defined over the doubled space. The linear section condition (2.5), i.e.,∂ m f = 0 , then can be interpreted as the para-holomorphicity of any fields on the doubled space.
The Nijenhuis tensor associated with the almost para-complex structure K, can be expressed as 14) and the integrabilities of the both two distributions L * ± can be summarized as the integrability of K: N K = 0 . By using the polarization tensors, we can also split the tangent bundle as (2. 15) In this paper, we always assume that K is integrable (N K = 0), 2 and then by using the para-(anti-)holomorphic coordinates, distributions L + and L − are respectively spanned by ∂ m and ∂ m . Then, L + can be identified as the tangent bundles T F + of a d-dimensional space F + with coordinates x m , which we call the physical space. It can be realized asx m = c m (c m constant) in the para-(anti-)holomorphic coordinates. Similarly, the unphysical gauge orbits So far, we have constructed the doubled space as a para-complex manifold (M, K). Now let us also consider the metric η IJ . The standard assumption in DFT is that the metric η IJ has the form (2.2) in the para-(anti-)holomorphic coordinates. Then we can easily see that Eq. (2.4) and When this relation is satisfied, the pair (K, η) is called a para-Hermitian structure and the doubled space (M, K, η) equipped with a para-Hermitian structure is called a para-Hermitian manifold. On an arbitrary para-Hermitian manifold, we can define a natural 2-form field which is called the fundamental 2-form. By definition, this satisfies If this 2-form is closed dω = 0 , the para-Hermitian manifold is called a para-Kähler manifold.
Example 1. In the conventional DFT, we assume that the doubled space is a para-Kähler manifold with K = K (0) and ω = ω (0) ≡ η K (0) . We can also consider a deformation of K (0) by performing a B-transformation, After the deformation, L + is spanned by e m ≡ ∂ m − b mn∂ n and the linear section condition becomes∂ m = 0 (see [33] where this b mn was introduced to discuss finite transformations in DFT). In addition, the fundamental 2-form becomes This is still a para-Kähler structure because dω (b) = 0, which follows from db = 0 .
Example 2. Another non-trivial example is given by where π mn is a Poisson-tensor satisfying π m[n ∂ m π pq] = 0. In this case, we obtain 27) and this is para-Hermitian but not para-Kähler unless π mn is constant.

Canonical generalized Lie derivative
Let us consider diffeomorphisms generated by the standard Lie derivative which is a symmetry of the conventional DFT action. The usual diffeomorphisms on the physical space F + are rather contained in another local symmetry, which is known as the generalized diffeomorphism. When the para-complex structure is given by K = K (0) , this symmetry is generated by the generalized Lie derivativê In this sense, the generalized Lie derivative is a generalization of the usual Lie derivative. The para-complex structure K (0) is invariant under this transformation, but if the diffeomorphism parameter V I has a general form (V I ) = (v m ,ṽ m ), it is transformed as Thus the diffeomorphism parameterṽ m causes a deformation of the foliation. In general, under a finite generalized diffeomorphism, K (0) is generally mapped to K (b) given in Eq. (2.24).
If we consider a general K, an issue arises for£ V . For two vector fields V = Π + (V ) and W = Π + (W ) that are tangential to the physical space F + , we havê where T IJK = T [IJK] is defined as Then, although both V and W are restricted to be tangent vectors on the physical space, the generalized Lie derivative does not reduce to the usual Lie derivative on the physical space due to the second term in Eq. (2.33). This prompts us to consider a modification of the generalized Lie derivative. In fact, there is the unique generalized Lie derivative that satisfies [11] This is known as the canonical generalized Lie derivative and is defined bŷ Here, ∇ is called the canonical connection 4 This is called para-Hermitian because it is compatible with the para-Hermitian structure, The difference between the two generalized Lie derivatives is called the generalized torsion and we can easily show the property (2.35) by using Eq. (2.33). Accordingly, we consider that the gauge symmetry of DFT for a general foliation is generated by the canonical generalized Lie derivative. In the case of para-Kähler manifolds (where dω = 0), we can easily see that V by using the identity In this sense,£ V is a modest modification of the generalized Lie derivative.

Born geometry
In DFT, the dynamical metric and the Kalb-Ramond B-field are packaged into the generalized metric H IJ which satisfies (2.42) 4 A short computation shows that canonical connection can be also expressed as By choosing a frame where K = K (0) is realized, we parameterize the generalized metric as 5 and interpret g mn as the metric on the physical space.
The second condition of Eq. (2.42) shows that J I J ≡ H I J is an additional almost paracomplex structure on the doubled space. This is sometimes called the chiral structure because this matrix defines a chirality of the string (see section 2.4). If this chiral structure satisfies into the condition (2.44), we obtain B mn = 0 , which looks very strong. In order to relax the requirement, one may assume that K and ω have the form [10,11,20,29] in the duality frame where the generalized metric has the form (2.43). This allows us to satisfy Eq. (2.44) but the integrability N + = 0 is broken when the B-field is not closed. Even when the integrability N + = 0 is broken, as long as the integrability N − = 0 is satisfied, we can define the physical space as F − (which satisfies L − = T F − ), and we may consistently formulate the gauged Born sigma model [29].
In this paper, we do not require Eq. (2.44) and do not include the supergravity fields, such as the B-field, into the fundamental 2-form ω . Then, the generalized metric does not describe the Born geometry, but since the action studied in the next subsection has the same form as that of the gauged Born sigma model, we shall use the nomenclature, the Born sigma model.
As we explain in the next subsection, what we include into ω are the field strengths of the worldvolume gauge fields.

Born sigma model for string
Here we consider the string sigma model. We suppose that the doubled space is a para-Kähler manifold with the structure (η, ω (0) ), and consider the action Here, µ 1 is the string charge (or tension), H IJ (x) is a generalized metric satisfying the linear section condition (Π (0) − ) J I∇J = 0, and DX I is defined by The scalar field λ(σ) is an auxiliary field that determines the tension, and the gauge fields In the second term, we have defined (2.48) and the second term can be expanded as We note that the covariant derivative DX I is invariant under the gauge transformation We also note that K (F ) describes the foliation of the d-dimensional space that the string lives in. This is different from the foliation characterized by K (0) and the deviation is characterized by the field strength F 2 (σ). If there are several strings propagating on the physical space, each string can live in a different d-dimensional space and each foliation is determined dynamically. 6 Then, different strings observe the physical space from different angles.
The equations of motion for λ and the gauge fields C m give e λ = µ 1 (for µ 1 > 0) and By taking the Hodge dual of this equation, we obtain (Π According to this relation, J I J = H I J is called the chirality operator. If we consider a flat background, the equations of motion give dA I = 0 and we can fix the gauge symmetry (2.50) as A I = 0 . Then we obtain DX I = dX I and the on-shell value of the action becomes where we have truncated F 2 for the sake of comparison. This is precisely Hull's action [6,25], and the topological term dx m ∧ dx m (which comes from the ω term) plays an important role, for example, in the computation of the partition function [26].

Boundary condition and D-brane
Under the equations of motion, a variation of the action becomes In order to make the variational principle well-defined, we require the boundary condition where n a denotes the vector field normal to the boundary ∂Σ. By introducing a projection operator (π D ) m n which has only the diagonal elements with values 1 or 0 , we can impose the Dirichlet boundary condition as (π D ) m n δx n ∂Σ = 0 .
where (π N ) m n ≡ δ m n − (π D ) m n . By using the equations of motion the Neumann condition can be also expressed in the standard form By noting that the Dirichlet boundary condition can be also written as both the Dirichlet and the Neumann boundary conditions can be summarized as If we again consider a flat background, DX I can be gauge fixed to dX I and the boundary condition reduces to This can be interpreted as a generalized Dirichlet boundary condition in the doubled space [6] that extends the conventional one (2.62). Since there are d "+1" in the diagonal elements of Π (F ) D , regardless of the choice of the matrix π D , the string is always attached to the "generalized Dirichlet brane" which is a d-dimensional object in the doubled space. In particular, when this object behaves as a p-brane in the physical space (namely when the trace of π N is p + 1), this object is called a Dp-brane [6]. In this way, the double sigma model (or the Born sigma model) allows us to describe the Dp-brane with various values of p as a single d-dimensional object in the doubled space [6].

Exceptional space
Here, we consider an extension of the same idea to the U -duality-covariant formulation, known as the exceptional field theory (EFT) [36][37][38][39][40][41][42][43][44][45]. In the E n EFT, we introduce an exceptional space with local coordinates x I which transform in the R 1 -representation of the E n U -duality group. In the exceptional space, the section condition can be expressed as 7 (3.1) Here, η IJ;K is an intertwining operator (called the η-symbol 8 ) which connects a symmetric product of the R 1 -representation and the R 2 -representation (labeled by I, J , K, · · · ). An important difference from the DFT case (2.3) is that the R 2 -representation is not a singlet. 9 In order to satisfy the section condition (3.1), we again introduce a projector satisfying and assume that any fields f satisfy the linear section condition Additional conditions appear if we consider n ≥ 7 but they do not affect the discussion here. 8 In [46] it is denoted as f M 1 N 1 P 2 but here we follow the notation of [47,48]. 9 Another difference is that the construction of the covariant derivative satisfying∇I η = 0 is non-trivial in EFT due to the last index of η IJ ;K (see [49] for a discussion on a such connection in the SL(5) EFT). Here we restrict ourselves to coordinate systems where η IJ ;K is constant such that∇I is reduced to ∂I . In the following, we show the explicit form of the matrices K and ω IL;K , and after that we discuss brane actions. For convenience, we employ the following notation for multiple indices. For example, Aī p represents . The factorial is introduced in order to reduce the overcounting: The antisymmetrized Kronecker delta is defined as δ jp] and we also define δ In the multiple-index notation, the latter Kronecker delta is denoted as δī p jp ≡ j 1 ···jp , which satisfies δī p kp δk p jp = δī p jp . For example, for a 6-form F 6 and a 3-form F 3 , we have We also use a bracket notation, such as A [īp Bī q · · · Cī r ] ≡ δk plq ···mr i p+q+···+r Ak p Bl q · · · Cm r . In the standard notation, this corresponds to It is noted that this bracket does not always coincide with the standard one even for single indices

8)
whereī denotes a single indexī 1 . Accordingly, when the bracket is defined in the modified sense, we should keep the barī inside the bracket. Let us give an another example. For a p-form A p and a (q + 1)-form B q+1 , we obtain (3.10) These notations are useful to remove unimportant numerical factors from various expressions.

M-theory section
When we consider the M-theory section, we expand the R 1 -representation as 11) and the R 2 -representation as (η IJ;K ) = η IJ;k , η IJ;k 4 , · · · . (3.12) We note that the dimensions of these representations are finite for n ≤ 8, but they are infinite for n = 9, 10, 11. In addition to η IJ;K , we also introduce which has the same matrix form as η IJ;K although the position of the indices are upside down.
The explicit forms of η IJ;K are as follows: (3.14) Then, we can easily see that a matrix indeed satisfies the conditions (3.2). Then, we obtain 16) and the matrix forms of (ω IJ;K ) = ω IJ;k , ω IJ;k 4 , · · · become We note that, unlike the DFT case, they are not antisymmetric in general: only the matrix k is. However, they play an important role in the brane actions, and we consider they are natural generalizations of the fundamental 2-form ω (0) IJ in DFT. 10 Similar to the DFT case, we can consider a more general Π ± or K by acting U -duality transformations (which generalize the B-transformation) where F 3 ≡ da 2 and F 6 ≡ da 5 are arbitrary closed 3-and 6-form fields. For these, ω become

Type IIB section
When we consider the type IIB section, the R 1 -representation is decomposed as where m, n, p = 1, . . . , d (= n − 1) , α, β = 1, 2 andm p denotes the multiple index. The Their explicit forms are given as follows: Similar to the M-theory case, we can compute Π (F ) ± and K (F ) as follows: where F α 2 ≡ da α 1 , F 4 ≡ da 3 , and F α 6 ≡ da α 5 are arbitrary closed forms. It is noted that the trace of the projector (Π By using the almost product structure K (F ) , we obtain the matrices ω (F ) as follows:

Generalized Lie derivative in EFT
In the conventional formulation of EFT, the generalized Lie derivative is defined as [41] £ (0) where Y IJ KL is an invariant tensor (e.g., Y IJ KL = η IJ;I η KL;I for n ≤ 6). However, similar to the DFT case, if we consider a non-constant polarization Π ± , this may not satisfy the propertŷ We note that this kind of modified generalized Lie derivative has been studied in several contexts [49,51].
If we require that tangent vectors on the physical subspace are maximally isotropic, i.e., we can show£ which gives a constraint for the generalized torsion. For the consistent formulation of EFT, we may need additional conditions, but here we do not study in further details. Of course, when the generalized torsion vanishes, the conventional EFT is recovered.
In the following, we study brane actions by using two product structures K (0) and K (F ) with vanishing generalized torsions T I J K . The former defines the physical subspace and the supergravity fields satisfy (Π (0) − ) J I ∂ J = 0 . On the other hand, the latter describes the foliation associated with the brane. This is described by the closed forms, collectively denoted by F . They correspond to the field strengths of the standard worldvolume gauge fields and in general dynamical.

Brane actions in M-theory
We consider the action  42) whereJ runs over the R 1 -representation other than the physical directions denoted by {i} .
Namely, we have The (p − 1)-form q K (brane) represents the charge vector associated with each brane. In this section, we consider the M2/M5-brane, and their corresponding charge vectors are given by where µ p is a brane charge (or tension) and we have defined Note that, in the doubled space, the R 2 -representation is a singlet and the charge vector is just a constant q K (string) = µ 1 . Under this identification, the action (3.41) reproduces (2.46). The second term in the action (3.41) can be expanded as Then, as naturally expected from the invariance of the action (3.41) under the generalized Lie derivative, these actions are the same as the ones proposed in [30,31] (note that the first term of Eq. (3.46) corresponds to the topological term proposed in Eq. (9.1) of [52]). As was shown there, they are (classically) equivalent to the standard (bosonic) M2/M5-brane theories.
In [30,31], the discussion was restricted to n ≤ 7 , but such a restriction is not necessary.
If we consider n ≥ 8 , the matrix size of the generalized metric H IJ becomes bigger and it can be infinite dimensional. However, the number of the auxiliary fields also increases accordingly. Since the actions for the irrelevant auxiliary fields are always given by algebraic quadratic forms, after eliminating these, we obtain the brane actions that have the same as the one studied in n ≤ 7. The only difference is the range of the index i = 1, . . . , n , and by choosing n = 11 , the full (bosonic) M2/M5-brane worldvolume theory in the 11D spacetime is recovered.

Brane actions in type IIB theory
We can consider the same action also in type IIB theory Here, the R 1 -representation is decomposed as in Eq. (3.20), and we choose the polarization tensor Π (0) ± as given in Eqs. (3.25) and (3.26). The parameterization of the generalized metric H IJ (X) is given in Appendix A.2. The charge vectors associated with a (p, q) string, D3-brane, and a (p, q) 5-brane are respectively given by (see Appendix B) We can again expand the second term in the action as As was shown in Ref. [31], the string action reproduces the conventional one for the (p, q)string (for a string, a similar U -duality-covariant sigma model is also discussed in [52,53]). The actions for the D3-brane and the (p, q) 5-branes have not been studied there. By eliminating the auxiliary fields, we find that these actions reproduce the following Wess-Zumino terms: Apparently, they do not have the standard form. Indeed, the D3-brane action contains a doublet of the worldvolume gauge field strengths F α 2 , although in the standard formulation we introduce only one gauge field. However, this kind of Wess-Zumino term that contains a doublet has been studied in S-duality-covariant formulations [54][55][56][57][58][59]. Then, it will be possible that the proposed theory is equivalent to the standard one after imposing a certain duality relation to the doublet of the gauge fields. 11 We will leave the consistency check with the standard formulation for future work.

Boundary conditions
Unlike the doubled case, the boundary condition in the exceptional space is non-trivial.
Before discussing higher-dimensional objects, let us consider the case of the (p, q)-string in type IIB theory (see [60] for a related study), where a variation of the action becomes δS e.o.m. (π D ) m n δx n ∂Σ = 0 ⇔ (π D ) m n n a ǫ ab ∂ b x n ∂Σ = 0 , (3.56) the Neumann boundary condition becomes where (π N ) m n ≡ δ m n − (π D ) m n . The equations of motion for the auxiliary fields λ and A I give e λ = |q| , |q| ≡ q α m αβ q β , (3.58) where m αβ and L I J are matrices including only the supergravity fields (see Appendix A.2) and g mn denotes the Einstein-frame metric. The action is then reduced to For simplicity, if we consider a flat background with L I J = δ I J (i.e., with vanishing p-form potentials), the equations of motion for x m (σ) lead to dP I = 0 and DX I is a closed form.
Then we can realize DX I = dX I , and the Neumann boundary condition becomes Under this situation, we can combine the Dirichlet/Neumann boundary conditions as where Π D is a projection operator and e F is an element of the U -duality group. Unfortunately, some elements of the Dirichlet projector Π D (denoted by " * ") cannot be determined because the third or lower components of the generalized vector dX I identically vanish For n ≥ 4 , our analysis only gives the lower bound d D ≥ n − 1 . According to the analysis based on the supersymmetry, it is claimed that d D = 2 n−2 for n ≤ 7 [60]. This indicates that there exists an object with the co-dimension 2 n−2 , and it is interesting to study the effective theory of such a higher-dimensional object in the exceptional space.
Instead of a string, we can also consider a higher-dimensional object, where we face a where a 1-form θ is given by Then, the Dirichlet boundary condition is while the Neumann boundary condition is

Brane actions in Hamiltonian form
In this section, we present brane actions in Hamiltonian form and see that the almost product structure K again plays an important role there. For this purpose, we decompose the worldsheet coordinates into the temporal and the spatial directions as (x a ) = (τ, xā) and decompose the intrinsic metric as

String action
Let us consider the first-order string action [61] If we identify the fieldsg mn (x) and β mn (x) with the generalized metric H IJ as this action reproduces the standard string action after eliminating the auxiliary fields P m . 12 The action S 1st is not manifestly T -duality covariant, but as it is discussed in [17], we can manifest the symmetry as follows. We expand the 1-form P m as and then the first-order action becomes (4.5) 12 The action S1st is related to the action S (2.46) (with e λ = µ1) as where we have identified Pm with Dxm and have defined (HIJ ) ≡ 0 0 0g mn . As long asg mn is invertible, the equations of motion obtained from S and S1st are equivalent.
Eliminating the auxiliary field q m , we obtain the Hamiltonian action where we have definedÑ ≡ N/ √ h and We note that p m (σ) is the usual momentum that is canonical conjugate to x m (σ) . This action can be also expressed in a T -duality-manifest form as where D τ X I ≡ (ẋ m , q m ) and q m =g mn x ′n − (g β) m n p n . This reproduces Tseytlin action ) KJ again plays an important role. According to the manifest T -duality covariance, this Hamiltonian action can be applied to backgrounds where H mn or H mn is not invertible (see [62] where it is applied to non-relativistic theories).

Nambu sigma model for a p-brane
In the case of a general p-brane, a covariant action similar to Eq. (4.2) has not been known.
However, an extension of the action (4.5) is known as the (non-topological) Nambu sigma model [63] where we have defined Eliminating the auxiliary fields qī p , we obtain the Hamiltonian action (4.14) We can consider the membrane theory by choosing p = 2 . In particular for n ≤ 4 , we can understand the index I as that of the R 1 -representation, and then we can express the Hamiltonian action in an E n U -duality-invariant form Here, in order to manifest the covariance, we have introduced a total-derivative term that contains the gauge field F 3 (see [62] where this Hamiltonian action has been applied to nonrelativistic theories). The point we would like to stress is that the projector Π (F ) + or the product structure K (F ) again plays an important role, and it is a natural extension of the product structure studied in the context of the para-Hermitian geometry or the Born geometry. It is also noted that, for n ≥ 5 , this Hamiltonian action is not U -duality covariant, but still describes the standard membrane theory under the identification [63] Other brane theories (i.e., p = 2) can be also studied in a similar way, but the action (4.10) reproduces only a part of the bosonic action. For example, if we consider p = 5 , we will obtain the M5-brane action with A 3 = F 3 = F 6 = 0 . Thus, the action (4.10) needs to be modified in order to consider the full bosonic theory.

Conclusions
By following the recent proposals that the doubled space is naturally defined as the para-Hermitian manifold or the Born manifold, we have introduced two types of almost product structures in the exceptional space: one defines the M-theory section while the other defines the type IIB section. By using the almost product structures, we have defined ω in each section and proposed natural extensions of the Born sigma model. The obtained actions are the same as the ones studied in [30,31] and reproduce the standard worldvolume theories for M2-and M5-branes as well as the (p, q)-string in type IIB theory. We have also studied the Hamiltonian actions for the string and the membrane and observed that the product structure K (F ) again appears in the action.
In the doubled space, the para-complex structure K has played an important role in defining the physical subspace, and the section condition can be understood as the paraholomorphicity of the physical fields. Various mathematical structures of the doubled space have been studied in the literature, but the geometry of the exceptional space has been poorly understood. The analysis presented in this paper suggests that the proposed (almost) product structure K I J is a natural extension of the para-complex structure in the doubled space, and it will be useful to describe the exceptional geometry in a more general framework. Indeed, as discussed in section 3.3, under a general choice of the almost product structure K, we need to modify the generalized Lie derivative by using the generalized torsion associated with K.
This will lead to the modifications of the generalized curvature in the exceptional space, and it might be an important future task to establish the geometry of the exceptional space by using the almost product structure.
It is also interesting to investigate whether we can formulate the manifestly U -duality invariant action that reproduce all of the brane actions. In the present formulation, we fix the dimension of the worldvolume to be p+1 in advance, and it is impossible to realize other brane action with a different dimensionality by performing a U -duality transformation. However, according to the discussion given in [60] (as well as the discussion of the boundary condition given in this paper), a string can end on a N n -dimensional generalized Dirichlet brane 13 in the D n -dimensional E n exceptional space, where the pair {N n , D n } can be summarized as According to this proposal, when this brane has a (p+1)-dimensional overlap with the physical space, it is understood as the familiar p-brane (see [6] where this viewpoint was proposed in the context of the doubled space). Then, it might be possible to formulate the effective theory of such generalized Dirichlet brane which reproduces the standard brane actions through a certain procedure that reduces the worldvolume dimension. In the case of the doubled space, an effective Lagrangian that describes all of the Dp-brane in a unified manner has been formulated in [64] (see also [65,66] for relevant recent works) and it is interesting to extend that to the case of the exceptional space.

The antisymmetrization is normalized such that
j k ] . We also define δ i 1 ···i k j 1 ···j k ≡ k! δ i 1 ···i k j 1 ···j k . The usage of the multiple-index notationī p is explained in detail at the beginning of section 3.

A.1 M-theory
When we study M-theory, we decomposed the E n generators into the GL(n) generators K i j as well as the positive-/negative-level generators positive level {Rī 3 , Rī 6 , · · · } , negative level {Rī 3 , Rī 6 , · · · } . (A.1) By using these, we can construct the generalized metric H IJ (with the "natural weight" 0) as Here,Ĥ IJ is constructed by exponentiating the GL(n) generators as where an overall rescaling has been done. The twist matrix L I J is made by using the positivelevel generators, whose matrix representations in the R 1 -representation are as follows: The fields {g ij , A 3 , A 6 } are standard bosonic fields in 11D supergravity. 14 The generalized metric M IJ with the "weight" 0 is given by M IJ = |det(g ij )| 1 n−2 H IJ which is an element of the E n group and has the unit determinant.

B Charge vectors
In this appendix, we review the construction of the charge vector q I (brane) for the standard branes 16 [31]. In the R 2 -representation, there exists a component with (p−1) antisymmetrized indices that corresponds to a p-brane. The pure charge vectorq I (brane) is defined by putting In type IIB theory, those for a (p, q) string, D3-brane, and a (p, q) 5-brane arē Here, the string and the 5-brane behave as S-duality doublets, and we have introduced a vector q α , where (q α ) = (1, 0) corresponds to the fundamental string/D5-brane while (q α ) = (0, −1) corresponds to the D1-brane/NS5-brane. These pure charge vectors do not transform covariantly under the generalized Lie derivative (i.e., under the p-form gauge transformations).
In order to obtain covariant vectors, we need to multiply a twist matrix L as follows.
To construct the twist matrix, we need the matrix representations of the E n generators in the R 2 -representation (t α ) I J . They can obtained by using the invariance of η IJ;K (t α ) L I η LJ;K + (t α ) L J η IL;K + η IJ;L (t α ) L K = 0 . and in type IIB theory, we obtain These charge vectors transform covariantly as discussed in [31].