In double field theory (DFT) (Refs. [1–7]), for the purpose of the manifest -duality covariance, we consider a -dimensional doubled space with the generalized coordinates (). In order to make contact with the conventional supergravity in -dimensions, it is useful to decompose the generalized coordinates into the physical coordinates () and the dual coordinates ; . By introducing the -duality-invariant metric, (1.1) the consistency condition of DFT, the so-called the section condition, is expressed as (1.2)

Here, represents that (1.3) are satisfied for arbitrary fields or gauge parameters and . Under the section condition, the gauge algebra generated by the following generalized Lie derivative is closed: (1.4)

As a natural generalization of DFT, the exceptional field theories (EFTs) (Refs. [8–16]) have been formulated in a manifestly -duality covariant manner (see Refs. [17–20] for the initial attempts). In EFT, the generalized coordinates () are defined to transform in a fundamental representation, called the -representation (see Appendix A.2). The generalized Lie derivative is defined by (1.5) where the -tensor for each () is given as follows (Ref. [11]; see Refs. [21–23] for the generalized Lie derivative in the context of exceptional generalized geometry): (1.6)

Here, e.g., is the gamma matrix for the group and is the totally symmetric tensor intrinsic to the group (see Appendix B for the details of these -dependent tensors). The gauge algebra of the generalized diffeomorphism is closed if the following section conditions are satisfied (Refs. [11,22]): (1.7) where is the antisymmetric tensor intrinsic to the group. Under the section condition, all fields can depend on at most coordinates (see Ref. [24] for a proof in the EFT).

In the above conventional formulation, the -tensor and the section condition strongly depend on the dimension , and when we consider applications of the EFT, we need to specify the dimension explicitly. A hypothetical “underlying EFT” (or 11D EFT), which reproduces all EFTs () from simple truncations, has been proposed in Ref. [25], but the program has not been completed yet. In this paper, we investigate such uniform formulations from a different approach. In our approach, the -tensor is expressed in terms of [or ] tensors and tensors are not used. Accordingly, the truncation to lower can be easily performed.

The present paper is organized as follows. In Sect. 2, we introduce -symbols as a natural generalization of the -invariant metric in DFT and explain how the -symbols are related to branes in M-theory/type IIB theory. The -tensor is expressed by using the -symbols and the -tensor. In Sect. 3, we find the explicit form of the -symbols and the -tensor. In Sect. 4, we show the explicit form of the section condition and the generalized Lie derivative. In Sect. 5, we propose a new linear section equation, and show that it reproduces the known linear section equation (Ref. [11]) in the case of the EFT. Section 6 is devoted to conclusions and discussion.

The section condition in DFT has been proposed on the basis of the level-matching constraint in string sigma model (Refs. [1–3]), (2.1)

In the canonical formulation, the level-matching constraint, or the momentum constraint , can be expressed as (2.2) where are the conjugate momenta to and are the generalized momenta, (2.3)

By supposing that the operator (2.4) acts as the generator of the diffeomorphism along , we can roughly identify with , and the momentum constraint corresponds to the section condition in DFT, .

A similar consideration has been given for M-theory branes, in Refs. [26,27]. In the case of an M2-brane wrapped on a 4-torus, the momentum constraint (: index for spatial coordinates on the M2-brane) is rewritten as (2.5) where (2.6)

Again by supposing the generalized momenta to act as with , the momentum constraint is expressed as the section condition, (2.7)

Similarly, in the case of an M5-brane wrapped on a 5-torus, the momentum constraint, (), has been expressed in a bilinear form (see Ref. [27] for the details), (2.8) where the matrices and have the form (2.9)

Here, and are arbitrary constants and the section conditions can be decomposed into two parts, (2.10)

The former condition is the same as the section condition coming from the M2-brane and the latter is intrinsic to the M5-brane.

A similar consideration for a D-brane in type II string theory was made in Ref. [28] (though the -duality covariance is not manifest there), and the general rule we observe is that each -brane provides the corresponding -symbol and the associated section condition . In fact, a set of multiple indices with one dimension fewer in the spatial dimension of branes is known to form the string multiplet of group. The dimension of the string multiplet for each -duality group is given as follows (see Ref. [29] for a concise review): (2.11)

As is clear from the dimension, the string multiplet is the same as the -representation that determines the section condition (Ref. [29]; see also Appendix A.2). Now, it is natural to expect that each brane in the string multiplet provides a particular -symbol and the corresponding section condition, and the sum of all these section conditions is equivalent to the section condition in the EFT. We thus introduce the following set of -symbols associated with branes in the string multiplet in M-theory and type IIB theory: (2.12) where the multiple indices are totally antisymmetrized and the ranges of the indices are , , and . Each -symbol corresponds to a brane specified below the underbrace (see Ref. [29] and also Ref. [30] for the notation of exotic branes ). The ellipses are relevant only for the EFT with , which is not considered here.

The above set of -symbols would be essentially the same as the set of -symbols introduced in an “F-theory” (Refs. [31–35]).¹ There, the -symbols were introduced as the Clebsch–Gordan–Wigner coefficients connecting and the -representation, and the Virasoro-like constraint was expressed as (2.13)

The generalized Lie derivative was obtained from the Virasoro-like constraint, and by comparing with the generalized Lie derivative, the -tensor in EFT was expressed as (2.14) where the singlet constraint was introduced for from consistency with the EFT. The explicit form of the -symbol was found in Ref. [34] using a different convention from ours.

In this paper, instead of attempting to translate the -symbols found in Ref. [34] into our convention, we utilize the linear map considered in Ref. [30]. As has been well known (Refs. [12,37]), EFT can reproduce both M-theory and type IIB theory. Depending on which theory one has in mind, there are two natural parameterizations of the generalized coordinates: for M-theory and for type IIB theory. The linear map in Ref. [30] provides a relation between the two parameterizations: (2.15)

When we consider the linear map, we decompose the physical coordinates () for M-theory and () for type IIB theory as (2.16)

Here, in the M-theory side corresponds to the coordinate on the M-theory circle. If we adopt the type IIA picture (by compactifying the M-theory circle), the linear map corresponds to a single -duality along the or directions in type IIA/IIB theory. Indeed, in Ref. [30], it was shown that the linear map between two generalized metrics, (M-theory) and (type IIB theory), (2.17) precisely reproduces the well-known -duality transformation rules for supergravity fields. In this paper, we apply this linear map to -symbols in M-theory/type IIB theory.

To be more specific, following the convention used in Ref. [30], we parameterize the generalized coordinates as (2.18) where the coordinates other than the physical coordinates are winding coordinates associated with some branes specified below the underbrace and ellipses again are relevant only for the EFT with . In the above parameterized generalized coordinates for M-theory, we begin by considering two -symbols, (2.19) which are trivial extensions of the -symbols associated with M2-/M5-branes shown in Eq. (2.9). Under a compactification on the M-theory circle, an M2-brane becomes a D2-brane or an F-string in type IIA theory, and under a -duality, it can become a D1/D3-brane or an F-string. Correspondingly, under the linear map, the -symbol can be mapped to an -symbol, or , associated with a D3-brane or an F/D-string in type IIB theory. Similarly, the -symbol can be mapped to an -symbol, , , or , associated with a Kaluza–Klein monopole (KKM), an NS/D5-brane, or a D3-brane in type IIB theory. Repeating the linear map, we can find almost all of the -symbols described in Eq. (2.18). The only -symbols that cannot straightforwardly be obtained from the linear map are and , which correspond to 8-branes in M-theory and -branes in type IIB theory, respectively. These branes are not related to other branes described in Eq. (2.18) via -duality transformations, but they are related to each other. In fact, by requiring the or covariance in the M-theory or type IIB theory sides, they also can be determined completely. Then, we find the explicit form of all -symbols (and also the -tensor) and can construct the -tensor through Eq. (2.14).

We expect that, in the same manner as Refs. [26–28], all of the -symbols obtained in this paper will also be read off from the momentum constraint (i.e., Virasoro-like constraint) in worldvolume theories of branes appearing in Eq. (2.12), but we leave the task for future work, and here we will concentrate on the determination of the -symbols for the EFT ().

In this section, we begin by showing the explicit matrix form of -symbols in two generalized coordinates, and , associated with M-theory and type IIB theory, respectively. Their derivations are explained in Sect. 3.3. In Appendix B, we explain how to reproduce the known -tensors from our -symbols.

By using the obtained -symbols, the section condition can be expressed as follows (see Ref. [22] for a quite similar section condition and also Ref. [25] for a section condition in the “underlying EFT”): (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) where and are defined in Eq. (3.48). In particular, when we consider, e.g., the EFT, the above section conditions are truncated easily to get (4.7)

The section condition in the type IIB parameterization, , can also be rewritten in a similar manner, though we will not show this explicitly.

There are two well-known solutions to the section condition. One is the solution, called the M-theory section, where (4.8) are satisfied; namely, on the M-theory section, all fields depend only on the coordinates . The other solution is called the type IIB section, where (4.9) are satisfied. In the type IIB section, fields depend only on the coordinates (see Ref. [37] for the type IIB section in the EFT and also Refs. [12–14,16] for a higher EFT).

On the M-theory section, the generalized Lie derivative reduces to the exceptional Dorfman bracket (Refs. [21–23,38]). Indeed, by using the -tensor, (4.10) and the explicit form of the -symbols and the -tensor, we obtain (4.11) where and . In the last line, we have repeatedly used the Schouten-like identities such as (4.12) which are satisfied for . This result precisely matches with the known result (Refs. [21–23,38]).

For a gauge parameter of the form , where , the generalized Lie derivative becomes (4.13)

In fact, a condition, (4.14) is necessary for the closure of the gauge algebra (Ref. [11]), and for , it is indeed satisfied under the section condition (1.7) (Ref. [11]). Therefore, a gauge parameter of the form is a generalized Killing vector for an arbitrary . Moreover, with satisfying (4.15) is also a trivial generalized Killing vector (Ref. [14]), (4.16) where the identity (3.65) is used in the second equality and Eq. (4.15) is used in the last equality.

On the other hand, if we choose the type IIB section, the generalized Lie derivative takes the form (4.17)

Again, and are trivial gauge parameters.

In the DFT, the section condition is expressed as . This condition states that is restricted to a -dimensional maximal null subspace in the generalized tangent bundle. We can specify the maximal null subspace by introducing a set of independent generalized vectors () satisfying (5.1)

If we consider a particular solution to the first equation, , that takes the form (5.2) the second equation in Eq. (5.1) gives , which is the commonly used section to reproduce the usual supergravity from DFT. More generally, if the matrix is invertible, we can always realize by a redefinition of ; . Then, by introducing an antisymmetric tensor , the general solution to the first equation in (5.1) becomes (5.3) which is just an rotation of the generalized vector . For this general , the second equation in Eq. (5.1) becomes (5.4)

We can easily show that this leads to the section condition (5.5)

In fact, in the context of generalized geometry, essentially the same set of generalized vectors has been considered in Ref. [39] (see also Ref. [40]). There, the maximal null subspace has been called the Dirac manifold or the Dirac structure, and the set of generalized vectors has been called the basis representation of the Dirac structure. In addition, it has been shown that the Dirac structure can be characterized by an antisymmetric tensor, which is denoted by here. Alternatively, we can also characterize the Dirac structure by using a pure spinor (Ref. [41]).

A linear differential equation similar to Eq. (5.1), which reproduces the section condition, is called the linear section equation in Ref. [11]. There, a linear section equation in EFT for was proposed, but the equation strongly depends on the dimension and it becomes complicated for higher . Here, using the -symbols, we propose a simple linear section equation, and show that it is equivalent to the proposal of Ref. [11] for the EFT.

Our linear section equations take the form (5.6) where () is a set of generalized vectors satisfying the null conditions (5.7)

If we show all of the indices explicitly, the linear section equations in the M-theory/type IIB parameterization become (5.8)

The number of independent null generalized vectors depends on the choice of the section. As was shown in Ref. [24], cannot be greater than , but we can always choose or , which correspond to the M-theory section and the type IIB section, respectively. In fact, the M-theory section and the type IIB section can be described by the following set of null vectors, () and (): (5.9)

In the former case, and require . On the other hand, and are trivially satisfied. The remaining conditions, and , require . Therefore, describes the M-theory section where all fields depend only on . The quadratic section condition is trivially satisfied on this section. Similarly, in the latter case, we can easily show that all fields depend only on , and describes the type IIB section.

In order to describe a more general section, we can rotate the above canonical sections, and , by -duality transformations: (5.10)

Since and behave as the Clebsch–Gordan–Wigner coefficients, they satisfy (5.11) where and are certain -duality-transformation matrices in the -representation. Moreover, the -tensor is invariant under the -duality transformations. Then, the transformed generalized vector, or , also satisfies the null conditions (5.12) which are the required properties for writing the linear section equations. In fact, the linear section equations specified by the transformed or lead to the quadratic section condition or . Indeed, the linear section equations require that the dual components of the transformed derivatives or vanish, and from this, we can easily show and or and , which are equivalent to the quadratic section condition.

In an example of the EFT, the null conditions for the generalized vectors, are (5.13)

If is invertible, we can choose and then the first equation requires . Then, we can express by using a 3-vector (i.e., antisymmetric third-rank tensor) , , and the second equation becomes (5.14)

This leads to (5.15) where is an arbitrary 6-vector. The last equation in Eq. (5.13) is trivially satisfied. Therefore, the most general parameterization is given by (5.16)

In this way, when is invertible, the most general parameterization of is obtained from via a -duality transformation generated only by negative-root generators and ( generators are not necessary).² It is also the case for lower exceptional groups . The same will be the case for also, and in that case, will be specified by parameters and . Similarly, in the case of the type IIB section, if is invertible, the most general parameterization of will be given by (5.17)

In the following, we show that our linear section reproduces the known linear section equation in the EFT, both for the M-theory and the type IIB sections.

In this paper, we obtained a set of -symbols associated with branes in the string multiplet, and reproduced the known -tensor in EFT with . Our expression does not depend on the tensors for a particular , and a reduction to lower can be easily performed. Using the -symbols (and the -tensor), we proposed linear section equations that reproduce the usual quadratic section condition. Equivalence to the known linear section for the M-theory and the type IIB sections in the EFT are shown.

Our considerations are limited to , but we can also consider the EFT, where the number of -symbols will be the same as the dimension of the -representation of , namely . According to Ref. [29], the branes in the string multiplet can be summarized as in Table 1. There, each brane in the table is wrapping a certain cycle in the 8-torus and behaves as a string with a tension in the uncompactified spacetime. We call the brane a “-brane” if the tension of the string takes the form

Table 1.

M-theory branes in the string multiplet for the EFT. In each column, all of the indices must be different. Branes in the left and right columns are dual to each other (Ref. [29]).

(6.1) where denotes the radius along the -direction (), and is the 11-dimensional Planck length. We also define and . It will be interesting to determine all of the -symbols associated with the branes.

In this paper, we have not discussed the role of the -symbols in worldvolume theories in detail, but in fact, they play an important role. In the -duality manifest formulation of the string, the equations of motion can be expressed as the self-duality relation (Ref. [44]) (6.2) where is the Hodge star operator on the worldsheet associated with the metric , and (6.3)

As a generalization of this relation, if we consider a membrane theory in the approach of Ref. [45], the equations of motion can be expressed as (6.4) by using the -symbol associated with an M2-brane. It will be interesting to see whether this kind of self-duality relation is satisfied for all of the branes in the string multiplet. It is also interesting to see how the -tensor appears in the brane worldvolume theories.