Exceptional M-brane sigma models and $\eta$-symbols

We develop the M-brane actions proposed in arXiv:1607.04265 by using $\eta$-symbols determined in arXiv:1708.06342. Introducing $\eta$-forms that are defined with the $\eta$-symbols, we present U-duality-covariant M-brane actions which describe the known brane worldvolume theories for M$p$-branes with $p=0,2,5$. We show that the self-duality relation known in the double sigma model is naturally generalized to M-branes. In particular, for an M5-brane, the self-duality relation is nontrivially realized, where the Hodge star operator is defined with the familiar M5-brane metric while the $\eta$-form contains the self-dual 3-form field strength. The action for a Kaluza-Klein monopole is also partially reproduced. Moreover, we explain how to treat type IIB branes in our general formalism. As a demonstration, we reproduce the known action for a $(p,q)$-string.


Introduction
String theory compactified on a d-torus has the O(d, d) T -duality symmetry, but the duality is not manifest in the conventional formulation. A T -duality manifest formulation for strings, called the double sigma model (DSM), was originally developed in [1][2][3][4][5][6], where the dimensions of the target spacetime are doubled by introducing the dual winding coordinates. Utilizing the idea of the doubled spacetime, a manifestly T -duality-covariant formulation of low-energy superstrings was developed in [4,[7][8][9][10], which is nowadays known as the double field theory (DFT). More recent studies on the DSM include [11][12][13][14]. Other than the fundamental string, higher-dimensional objects also transform covariantly under T -duality. A T -duality-covariant action for D-branes was constructed in [15] (see also [5]) and a covariant action for a family of type II 5-branes [i.e. NS5-brane, Kaluza-Klein Monopole (KKM), and the exotic 5 2 2 -brane] was constructed in [16].
In this paper, we develop the worldvolume theories for M-branes proposed in [49]. The proposed theory is based on the geometry of the exceptional spacetime (introduced in EFT) and can reproduce the conventional worldvolume theories for the M2-brane and M5-brane in a uniform manner. The action for an Mp-brane takes the form (1.1) However, the U -duality covariance has not been manifest in Ω p+1 . In this paper, by using the η-symbols recently determined in [50], we introduce a covariant object η IJ , to be called the η-form, and propose a duality-covariant action that reproduces the above action. As we shall argue later, the η-form can be regarded as a natural generalization of the O(d, d)-invariant metric η IJ in DFT or DSM. Indeed, we show that the self-duality relation in DSM, can be naturally generalized to η IJ ∧ P J = M IJ * γ P J (1. 3) for an Mp-brane. Moreover, we argue that the action for a KKM can also be naturally reproduced in our formalism, although the whole action is not reproduced due to limitations of our analysis. We also demonstrate that our formalism can reproduce brane actions for type IIB branes.
The present paper is organized as follows. In Sect. 2, we briefly review the DSM constructed in [12] and explain a slight difference from our approach. In Sect. 3, we apply our approach to M-branes; M0, M2, M5-branes and KKM. In Sect. 4, we explain how to apply our formalism to type IIB branes and reproduce the action for a (p, q)-string. A possible application to exotic branes is discussed in Sect. 5. Section 6 is devoted to conclusions and discussion.

Double sigma model
In this section, we review the standard construction of the DSM and explain a slight difference from our approach. The difference is not significant in the DSM, but it becomes important when we consider higher dimensional objects in the following sections.

A brief review of double sigma model
Let us begin with a brief review of Lee and Park's DSM [12] (known as the string sigma model on the doubled-yet-gauged spacetime). The action takes the form where η IJ is the O(d, d)-invariant metric, γ ab (σ) is the intrinsic metric on the worldsheet, X I (σ) is the embedding function of the string into the doubled spacetime, and H IJ (X) is the generalized metric satisfying the section condition ∂ K ∂ K H IJ = 0 . According to the section condition (or equivalently the coordinate gauge symmetry [51]), there are d generalized Killing vectors, which take the form∂ i (i = 1, . . . , d) when H IJ depends only on the x i coordinates.
Associated to the isometries, we introduce 1-form gauge fields A I (σ) satisfying for an arbitrary supergravity field T (x) , and define the covariant derivative DX I (σ) ≡ dX I (σ) − A I (σ) .
In order to see the equivalence to the conventional string sigma model, let us consider a duality frame where∂ k H IJ = 0 is realized. In such frame, (2.2) requires A I and DX I to have the following form: , By further using the parameterization of the generalized metric the action becomes Eliminating the gauge fields A i , we obtain the action which is the familiar sigma model action for the bosonic string up to a total-derivative term.
The DSM is thus classically equivalent to the conventional string sigma model. The action It is also invariant under generalized diffeomorphisms in the target doubled spacetime [12].

Our approach
In this paper, we basically follow the approach of Lee and Park, but there are slight differences.
Following [50], we introduce a set of null generalized vectors λ a (a = 1, . . . , d) satisfying These λ a specify a solution of the section condition, and an arbitrary supergravity field T (x) must satisfy the linear section equation [50] λ a I η IJ ∂ J T (x) = 0 . This is a minor difference (though it becomes important when we consider brane actions).
A major difference is in the parameterization of fluctuations. In the DSMs known in the literature, fluctuations of a string are described by the embedding function X I (σ), but we take a different approach, which is important in the generalization to branes. We first choose the section (2.9), where all fields and gauge parameters depend only on the physical coordinates where the bar represents that the string is static. If we introduce a 1-form it corresponds to dX I (σ) of a string in the static gauge, X 0 (σ) = σ 0 and X 1 (σ) = σ 1 . In order to describe a fluctuation of the string, we perform a finite active diffeomorphism along a gauge parameter ξ I (x) = (ξ i ,ξ i ) satisfying∂ i ξ I = 0 (see Figure 1). Under the section (2.9), a generalized diffeomorphism e£ ξ can be decomposed into a B-field gauge transformation and a usual diffeomorphism  14) and the B-field gauge transformation further maps it as .
We thus introduce a generalized vector E I (σ), which describes fluctuations of a string, as , (2.16) where the prime has been removed for simplicity. The scalar fields X i describe fluctuations of a string inside the d-dimensional physical subspace of the doubled target space (with coordinates x i ), while the 1-form A i describes the fluctuation along the dual directions in the doubled spacetime. In general, since the integrability condition is violated, i.e.
we cannot find the embedding functions X I (σ) that realize E I (σ) = dX I (σ) . However, inside the physical subspace, the integrability condition, (∂ τ E σ − ∂ σ E τ ) i = 0, is satisfied and the worldsheet is a manifold described by X i (σ) as usual. Thus, the violation in the dual components may be related to the gerbe structure discussed in [52]. In this paper, instead of assuming the existence of the embedding functions X I (σ), we parameterize fluctuations of a string by using the diffeomorphism parameters ξ I , or equivalently {X i (σ), A i X(σ) }.
Since E I is obtained by acting a generalized diffeomorphism on a generalized vectorĒ I , E I also transforms as a generalized vector. Such behavior of E I is ensured as long as F ij transforms (like the B-field) as under an infinitesimal diffeomorphism. It should also be noted that E I is a null generalized vector; η IJ E I E J = 0. Assuming the null property, our parameterization (2.16) is the most general parameterization up to duality rotations. Now, our action is given by In the duality frame (2.9), the condition (2.10) leads to 20) and, in the following, we consider P i as the fundamental variable rather than A i . If we rewrite the action as we observe that F ij appears only in the second term. In the second term, since the only quantity with an upper index is dX i (other than the Kronecker delta), we see that F ij appears only through the pullback, Indeed, we can explicitly expand the second term as Therefore, the fundamental fields in our action are Namely, not all components of A i X(σ) appear in the action-only the pullback A 1 (σ) does.
Eliminating the auxiliary fields P i (σ) by using their equations of motion, we obtain For later convenience, let us also comment on the self-dual relation [1,5,6,12]. The equations of motion for the auxiliary fields can be written as A duality-covariant rewriting of this equation is known as the self-dual relation, and takes the form In this paper, we find a similar self-dual relation for M-branes that determines all of the auxiliary fields in terms of the conventional fields.

M-branes in exceptional spacetime
In this section, we consider worldvolume actions for M-branes. We decompose the elevendimensional spacetime into an (11 − d)-dimensional "external space" and a d-dimensional "internal space," and enlarge the internal space into an exceptional space with dimension

Action for an Mp-brane
In order to describe M-branes, we parameterize the generalized coordinates in the E d(d) exceptional spacetime (d ≤ 7) as where i = 1, . . . , d and I = 1, . . . , dim R 1 . We also parameterize the generalized metric M IJ as follows by using the fields in the eleven-dimensional supergravity [22,37,53]: Here, (G ij , C i 1 i 2 i 3 , C i 1 ···i 6 ) are the conventional fields in the eleven-dimensional supergravity, |G| ≡ det(G ij ), and (R i 1 i 2 i 3 ) I J and (R i 1 ···i 6 ) I J are E d(d) generators in the R 1 -representation (see Appendix A.2 for the details). We also defined δ Similar to the case of the DSM, we specify the section by introducing a set of null generalized vectors λ a I (a = 1, . . . , d) satisfying [50] λ a I η IJ; where the explicit forms of η IJ; I and Ω IJ are given in Appendix A.3. For a given λ a I , the linear section equations for arbitrary supergravity fields and gauge parameters T (x) become Using the same λ a , we can express a condition for A I as which corresponds to (2.10) in the DSM. For a natural choice of λ a , supergravity fields depend only on the physical coordinates x i and A I takes the form Similar to the DSM, we describe fluctuations of a p-brane by using the 1-form-valued null generalized vector E I (σ) . In the case of the exceptional sigma model, we parameterize the null generalized vector as where we defined (3.13) As in the string case, X i , A i 1 i 2 , and A i 1 ···i 5 are understood as functions of the diffeomorphism parameters ξ I that fluctuate a static brane. In order for E I to transform as a generalized vector, F i 1 i 2 i 3 and F i 1 ···i 6 should transform as under an infinitesimal generalized diffeomorphism along . Now, we define the generalized vector P I (σ) as and regard {P i 1 i 2 , P i 1 ···i 5 , P i 1 ···i 7 , i } as the fundamental fields instead of Unlike the doubled case, the η-symbols η IJ; I in the E d(d) exceptional spacetime contain an additional index I [50]. Then, in order to describe a p-brane, we introduce a (p − 1)-form Q I that transforms in the R 2 -representation, and define a (p − 1)-form-valued η-symbol which we call the η-form. In particular, when we consider an Mp-brane (p = 0, 2, 5), we choose Q I as follows: µ p are constants representing the brane charge, and we have introduced an abbreviated notation Then, we propose the following actions: where η (Mp) IJ ≡ η IJ; I Q I (Mp) . Note that the M5-brane charge (3.17) has been obtained from the static "purely M5-brane charge"Q I (M5) through the active generalized diffeomorphism (3.12), where the transformation matrix L I J for the R 2 -representation is given by (see Appendix A.2) The M2-brane charge is invariant under the active diffeomorphism, L I J Q J (M2) = Q I (M2) . As long as F i 1 i 2 i 3 and F i 1 ···i 6 behave as in (3.14), Q I (Mp) transforms as a generalized vector in the R 2 -representation and hence the η-form η IJ transforms as a generalized tensor.
In our actions, the generalized metric M IJ (X) includes an overall factor |G(X)| 1 9−d , which is important for the duality covariance in EFT. However, it does not play an important role in the worldvolume theory because it can be absorbed into the intrinsic metric γ ab . For convenience, we introduce an independent scalar field eω (σ) inside M IJ (X) and regard the combination, e ω(σ) ≡ eω (σ) |G(X)| 1 9−d , as a new fundamental field. Namely, in the following, when we denote M IJ (X) in the worldvolume action, it means and e ω(σ) is an independent field. For a p-brane (p = 1), the action has a local symmetry, and e ω(σ) is not a dynamical field. Indeed, as we see later, e ω disappears from the action after eliminating γ ab by using the equation of motion. Only for the case of a string (p = 1) in type IIB theory does the new scalar field e ω play an important role (see Sect. 4.1).
In addition, the generalized vector E I contains quantities like F i 1 i 2 j dX j and F i 1 ···i 5 j dX j . As we explained in the doubled case, since all of the indices of F i 1 ···i p+1 are contracted with dX i in the action, only their pullbacks can appear in the action. Then, from the dimensionality, for example, F 6 cannot appear in the M2-brane action, and the fundamental fields can be summarized as follows: Our action for an Mp-brane (p = 0, 2, 5) can be summarized as This action is manifestly invariant under a generalized diffeomorphism along V I , where v i is restricted to be tangent to the worldvolume and we have defined The covariance of our action under global U -duality rotations is discussed in Sect. 3.6.
In order to expand the action explicitly, it is convenient to define the untwisted vector Then, we can expand the first term of the action as We can also calculate the second term of the action as where ∧ E J expressed in the above forms are the same as Ω 2 and Ω 5 introduced in [49] (up to conventions), and the actions presented above can be understood as a rewriting of the actions in [49] making the duality covariance manifest.
For later convenience, we also define

M0-brane
Let us consider the simplest example, the action for a particle in M-theory, sometimes called the M0-brane. The action is simply given by (see also [54,55] for particle actions in extended spacetimes) The equations of motion for the auxiliary fields P i 1 i 2 , P i 1 ···i 5 , and P i 1 ···i 7 , i givê and by eliminating the auxiliary fields, we obtain where v ≡ e −ω |γ τ τ | γ τ τ . By considering v as the fundamental variable (instead of the redundantly introduced fields ω and γ τ τ ), this is the bosonic part of the superparticle action discussed in [56].

Type IIA branes: D0-brane
For completeness, we review how to reproduce the D0-brane action from the above particle action [56]. By considering the reduction ansatz where r, s = 1, . . . , d − 1 and x M represents the M-theory direction, the action (3.37) becomes From the equations of motion for X M , we obtain where µ is the integration constant, and using this, the action becomes Here, we have added a total-derivative term µ ∂ τ X M . Using the equation of motion for v, we obtain the standard D0-brane action

M2-brane
Let us next consider the action for an M2-brane We derive the conventional action for the usual fields X i by using the equations of motion for auxiliary fields P i 1 ···i 3 , P i 1 ···i 5 , P i 1 ···i 7 , i , and γ ab . The equations of motion for P i 1 ···i 5 and P i 1 ···i 7 , i can be written asP Using these, the equation of motion for P i 1 i 2 becomes These equations of motion completely determineP I andẐ I in terms of X i and γ ab , The intrinsic metric γ ab can also be determined by using its equation of motion, Indeed, from this and the above solutions forP I , we obtain This leads to a is satisfied at an initial time, it must be always satisfied, namely (3.51) Using the above equations of motion, the action for X i becomes This is the bosonic part of the well-known membrane action [57].
Now, let us show the self-duality relation. Using the equations of motion, we can show This relation straightforwardly leads to the self-duality relation

Type IIA branes: D2-brane and F-string
For completeness, let us review the derivation of the actions for a D2-brane and a fundamental string from the M2-brane action. In order to obtain the D2-brane action, we follow the procedure of [56]. We first rewrite the action (3.52) as by introducing an auxiliary field v . Under the reduction ansatz the action becomes where Y 1 ≡ dX M + C 1 and we used the identity By introducing a Lagrange multiplier A 1 that imposes the constraint dY 1 ≡ dC 1 , we can rewrite the action as where we defined F 2 ≡ dA 1 and F 2 ≡ dA 1 − B 2 , and Y 1 is regarded as a fundamental field.
By eliminating Y a and using we obtain Finally, using the equation of motion for v, we obtain the well-known D2-brane action On the other hand, when we derive the string action, we first make an ansatz, Then, we can easily reproduce the Nambu-Goto-type action for a fundamental string

M5-brane
Let us next consider an M5-brane action where The equations of motion for P i 1 ···i 7 , i and P i 1 ···i 5 givê From these, the equation of motion for P i 1 i 2 takes the following form: Then, the equations of motion for auxiliary fields can be summarized as It should be noted that if we compute Z I = (L T ) I JẐ J for d ≤ 6 as its time component appears to be reproducing the generalized momenta, Eq. (2.9) in [58], obtained in the Hamiltonian analysis. The equation of motion for γ ab and the above solution where we have defined h a 1 ···an, b 1 ···bn ≡ h a 1 c 1 · · · h ancn δ b 1 ···bn c 1 ···cn and By rewriting this as and taking the square root, we obtain or Here and hereafter, we raise or lower the worldvolume indices a, b by using the induced metric h ab . The trace of (3.73) gives 75) and the action becomes This is the action obtained in [49], and, as was shown there, at least in the weak field limit |H 3 | 1, this theory is equivalent to the conventional M5-brane theory. In the following, we will check the equivalence at the non-linear level.

Results from the superembedding approach
A variation of our action (3.76) with respect to A 2 becomes (up to a boundary term) By using the covariant derivative D a associated with h ab , the equation of motion becomes This is consistent with the non-linear self-duality relation [49] although the self-duality relation cannot be derived from our action. On the other hand, under a simultaneous variation, δX i = v i and δA 2 = 1 2! v i C ij 1 j 2 dX j 1 j 2 (see [59]), the action (3.76) changes (up to a boundary term) as where we have defined Namely, we obtain the equations of motion In order to evaluate the last term, we recall the invariance of the action under a worldvolume where a boundary term is neglected because it is irrelevant. Since the diffeomorphism parameter ξ a (σ) is arbitrary, we obtain [59] Using this identity and the non-linear self-duality relation (3.80), we can express the equations of motion (3.83) as where we defined a projection, In order to compare the above equations of motion with the known ones, let us review the familiar results obtained in the superembedding approach [60,61]. In the superembedding approach, we introduce a self-dual 3-form field satisfying We then define An important relation that relates h abc and the 3-form field The above quantities satisfy [60] (3.91) We also define and then we can show the following nontrivial relations [62,63]: (3.93) If we introduce the 5-brane co-metric as [59][60][61][62] it satisfies the following relations [62,63]: Note that the 5-brane co-metric is proportional to the open membrane co-metric studied in [64,65]. Using the co-metric, we can express the non-linear self-duality relation for H 3 The equations of motion for scalar fields are obtained as [60,61] From the relations (3.93), we can easily see and the known non-linear self-duality relation (3.96) is equivalent to our relation (3.80). We can also show the nontrivial relation which takes the same form as the action studied in [59,66].

Self-duality relation for M5-brane
In this subsection, we show the self-duality relation for the M5-brane Using the equations of motion, the left-hand side becomes (3.104) Then, our task is to show that this generalized vector is equal to The nontrivial relations are the first and the second rows, (3.106) We show that we can simplify N 1 as Similarly, N 2 becomes Finally, N 3 becomes In this way, we have shown the nontrivial self-duality relation for the M5-brane.

Action for a Kaluza-Klein Monopole
As the last example, let us consider a KKM in M-theory. In fact, a KKM couples to the mixed-symmetry potential C i 1 ···i 8 , j , but this potential appears in the generalized metric The main difference from the previously considered M-branes is that a KKM requires the existence of an isometry direction generated by a generalized Killing vector k I . In this case, employing the standard procedure in the gauged sigma model, we introduce an additional 1-form gauge field a 1 (σ) and include it in A I , In other words, the generalized vector P I is modified as Supposing that the generalized Killing vector takes the form k I = (k i , 0, . . . , 0), we have We then consider the action where η (KKM) IJ takes the following form by neglecting the gauge fields: More explicitly, we consider the following action: Since we are neglecting the background gauge fields, the first term simply becomes The equation of motion for P i 1 ···i 7 , i gives and the equations of motion for P i 1 ···i 5 and P i 1 i 2 give Using these, the equation of motion for γ ab becomes the above equation can be expressed as and we obtain This leads to (3.126) and we finally obtain In this way, we can reproduce the well-known action for a KKM.
In order to introduce the worldvolume gauge fields, we need to modify the η-form, In the E 7(7) case, we cannot consider exotic branes since there are no winding coordinates (or auxiliary fields P i 1 ···i 8 , j 1 j 2 j 3 and P i 1 ···i 8 , j 1 ···j 6 ) for these branes. However, in the E 8(8) case, we can consider similar actions like IJ ∧ E J , although the explicit forms of the η-symbols, η IJ , are not yet determined. In the E 8(8) case, the generalized metric does not contain the potentials C i 1 ···i 9 , i 1 i 2 i 3 and C i 1 ···i 9 , i 1 ···i 6 that couple to the exotic 5 3 -brane and the 2 6 -brane, but we can consider the truncated action like the KKM action presented in this subsection. In order to reproduce the whole action for a 5 3 -brane and a 2 6 -brane, we are led to consider the E 9(9) exceptional spacetime. Another possibility to describe a KKM or exotic branes in d ≤ 7 is discussed in Sect. 5.

Comments on duality symmetry
In the previous sections, we have discussed our sigma model actions only in the usual section, where the set of null vectors λ a take the simple form, (λ a I ) = (δ a i , 0, . . . , 0). In such cases, E I , A I , and P I transform covariantly under generalized diffeomorphisms (which do not change the section λ a ), and our action was manifestly invariant. Since a subgroup of the T -or Uduality group, known as the geometric subgroup, can be realized as a rigid part of generalized diffeomorphisms, invariance of our action under the geometric subgroup is also manifest. In this subsection, we consider global duality transformations that change the section λ a , and show that E I , A I , and P I transform covariantly. In the conventional formulation of string theory/M-theory, such duality symmetry exists only in constant background, and we assume here that the supergravity fields are constant (unless otherwise stated).

Obstacle to manifest U -duality covariance
Let us begin with a brief review of the obstacle to describing the equations of motion in a manifestly duality-covariant form [46,47].
In the DSM defined in a constant background, the equation of motion for P i gives and taking the exterior derivative, we obtain where we used the equation of motion for X i in the last equality. Namely, for a given solution, we can (at least locally) findX i (σ) that satisfies P i = dX i . Then, we can express P I as P I = dX I , where (X I ) ≡ (X i ,X i ), and the equations of motion become  Remarkably, unlike the case of the DSM, this does not mean dP I = 0 . Indeed, as was pointed out in [47], if we consider a solution of an M2-brane (for d ≥ 4) {X i } = {σ 0 , α σ 1 cos(ω σ 0 ), α σ 1 sin(ω σ 0 ), β σ 2 , 0, . . . , 0} (α, β, ω : constant) , (3.135) we find that dP i 1 i 2 = 0 although (3.134) is satisfied. The only exception is the M2-brane in d = 3, called the topological membrane [47]. In that case, the equations of motion give where ε ijk ≡ √ −G ijk , and dP I = 0 is automatically satisfied. Then, at least locally, we can find the dual coordinates Y i 1 i 2 satisfying P i 1 i 2 = dY i 1 i 2 and the self-duality equation becomes This is covariant under the whole U -duality group E 3(3) = SL(3) × SL(2) [47]. In general cases with d ≥ 4, although we cannot express P i 1 i 2 as P i 1 i 2 = dY i 1 i 2 , the self-duality relation is itself still satisfied, and it is formally covariant under E d(d) transformations In particular, under global U -duality transformations generated by R i 1 i 2 i 3 and R i 1 ···i 6 , which we call the ω-transformations, P I is transformed as and, for example in d = 4, we have The problem discussed in [47] is basically that if we continue to use the parameterization the non-closedness dP i 1 i 2 = 0 leads to the non-integrability of dX i Then, the conclusion of [47] was that ω-transformations are not allowed and (3.138) is covariant only under the "geometric subgroup" generated by {K i j , R i 1 i 2 i 3 , R i 1 ···i 6 } (i.e., coordinate transformations GL(d) and constant shift of C 3 and C 6 ). In the following, we stress that the parameterization (P I ) = (dX i , 2! ) should be changed under ω-transformations, and the integrability condition dP i = 0 should be modified as d λ a I P I = 0 , (3.144) which is important to allow for the whole duality symmetry.

Duality covariance
Let us consider the DSM, where the duality group is O Here, the K i j correspond to general coordinate transformations GL(d) and the R ij = R [ij] correspond to the B-field gauge transformations, and these generate the geometric subgroup.
The correspondents of the ω-transformations, which change the section λ a , are called βtransformations that are generated by the remaining generators R ij = R [ij] .
In the following, we show that E I (σ) transforms covariantly where T (x) represents a supergravity field or a diffeomorphism parameter in the doubled spacetime. Originally, E I was defined as E I = e£ ξĒ I by using the staticĒ I defined in (2.12).
In the β-rotated frame,Ē I and the diffeomorphism parameter ξ I take the form In addition, the structure of the generalized diffeomorphism is also different according to the change of the section. By employing a convention, where∂ i is replaced by β ij ∂ j due to the linear section equations (3.148), a derivative in the β-rotated frame becomes Then, the generalized Lie derivative of an arbitrary generalized vector W I becomeŝ , (3.152) and we can show that a finite generalized diffeomorphism takes the form, which is precisely the β-rotated version of (2.13) [where the usual diffeomorphism and the B-field gauge transformation have precisely the same form as (2.13)]. Then, the components of E I described in the β-rotated frame become and dX i can be extracted from P I as dX a = λ a I P I by using λ a I . Therefore, the correct integrability condition (or the Bianchi identity) to require is d(λ a I P I ) = 0 as advocated in (3.144). Note also that if the original background is not constant, the generalized metric after the β-transformation includes the dual-coordinate dependence from (3.148). Since scalar fieldsX i (σ) are not introduced in our DSM, we cannot define our DSM in such background. This is the reason why we have supposed the background to be constant. Of course, since the supergravity fields are functions only of x i ≡ x i − β ijx j in the β-rotated background, instead of X i (σ), we can introduce X i (σ) as the fundamental variables in our DSM, but it is equivalent to going back to the usual section (λ a I ) = (δ a i , 0). We can straightforwardly also apply the above discussion to M-brane sigma models. For example, in the case of d = 4 discussed around Eq. (3.141), the ω-transformation rotates the usual section λ a I = (δ a i , 0) as There, the linear section equations (3.8) show∂ ij = ω ijk ∂ k , and a derivative becomes Accordingly, the generalized Lie derivative becomeŝ Then, components of E I described in the ω-rotated frame become and we see that E I transforms covariantly under the ω-transformation. Moreover, the correct parameterization of P I in the ω-rotated frame is Even for the higher-dimensional case d ≥ 5, from a similar argument, it will be possible to show that E I transforms covariantly, as is clear from the construction.

On dual coordinates
For completeness, we also comment on a section λ a ≡ (λ aI ) = (0, δ i a ), where supergravity fields depend only on the dual coordinatesx i . On this section, generalized diffeomorphisms are combinations of the usual Lie derivative (with opposite indices) and β-transformations.

(3.163)
In this case, A I and P I take the form The scalar fieldsX i describe fluctuations along the dual directions, while the V i describe fluctuations along the x i -directions. Our action then becomes where we defined d ≡ dσ a ∧ ∂ a (a = 1, 2) and V 1 (σ) ≡ V i X (σ) dX i (σ) is regarded as a fundamental variable. By parameterizing the generalized metric as and eliminating the auxiliary fields P i , we obtain the action which is the well-known dual action [1] if the background is constant. Again, note that the integrability condition becomes d(λ aI P I ) = d 2X a = 0 . We can also consider similar parameterizations of E I in the M-brane actions by choosing non-standard sections. Unlike the conventional DSM, our sigma model does not include all of the generalized coordinates X I (σ) as the fundamental variables, but we can choose a part of generalized coordinates depending on the choice of the section.

Type IIB branes in exceptional spacetime
In this section, we explain how to reproduce worldvolume actions for type IIB branes. The detailed analysis will be reported elsewhere, but here we explain the basic procedure and demonstrate that we can reproduce the action for a (p, q)-string. ) . (4.1) Each coordinate is the winding coordinate associated with the brane specified below. For d = 8, y i 1 ···i 7 , j includes 64 coordinates, and among these, 56 coordinates with j ∈ {i 1 , . . . , i 7 } correspond to the KKM while the remaining 8 coordinates with j ∈ {i 1 , . . . , i 7 } may correspond to 8-branes (known as M8-branes). If we decompose the physical coordinates x i as (x i ) = (x r , x M ) (r = 1, . . . , d − 1) where x M represents the M-theory direction, we can decompose the above generalized coordinates as those suitable for type IIA branes, , y r 1 r 2

(4.3)
In order to obtain the generalized coordinates for type IIB branes, we further decompose the physical coordinates in the type IIA side as (x r ) = (x a , x y ) (a = 1, . . . , d − 2) and perform a T -duality along the x y -direction. Under T -dualities, dependence of brane tensions on the string coupling constant g s does not change, and we summarize the mapping between the winding coordinates [53] in the following way. The type II branes with tension proportional to g 0 s are the fundamental string (F1) and the Kaluza-Klein momentum (P) while those with tension proportional to g −1 s are D-branes. By employing the convention of [53], their winding coordinates are mapped under the T -duality as follows: x a x a x y y 1 y y 1 a y 2 y −y 2 a y a 1 a 2 y y a 1 a 2 a 3 y 1 a 1 ···a 4 y y 1 The type II branes with tension proportional to g −2 s include the NS5-brane, KKM, and the exotic 5 2 2 -brane. Their winding coordinates are mapped as follows from type IIA theory to type IIB theory: y a 1 ···a 6 y,by w w y a 1 ···a 6 y,b 1 b 2 −y 2 a 1 ···a 4 y −y 2 a 1 ···a 5 y a 1 ···a 5 y,y y a 1 ···a 5 y,b ±y a 1 ···a 6 ,b ±y 1 a 1 ···a 6 y,by y 1 Here, the bar, as in y a 1 ···a 5 y,b represents thatb ∈ {a 1 , · · · , a 5 } and ± represents that the sign is not determined yet in [53].
There are another set of 7-branes that also have tension proportional to g −2 s but are not connected to other branes under T -dualities. The winding coordinates for the eight 7-branes in the type IIA side are y a 1 ···a 5 y,b (b ∈ {a 1 , · · · , a 5 }), y a 1 ···a 6 ,y , and y a 1 ···a 6 y . Although we have not identified their transformation rule yet, a natural expectation is as follows:  Here, the type IIB coordinates, y a 1 ···a 5 y,b (b ∈ {a 1 , · · · , a 5 }), y a 1 ···a 6 ,y , and y (12) a 1 ···a 6 y correspond to seven 7 2 -branes and a 7-brane that (together with the D7-brane and the 7 3 -brane) behaves as a triplet under SL(2) S-duality transformations. The detailed properties of these 7-branes are not well known, but they are necessary to construct a U -duality multiplet.
The type II branes with tension proportional to g It is interesting to note that, as has been uncovered in [69] (see also [70]), the T -duality transformation rules for the winding coordinates are very simple (up to the conventiondependent sign factor). For a type II brane with tension proportional to g −n s , if we consider the winding coordinate with m-number of y-indices, after a T -duality along the y-direction, we obtain a winding coordinate with (n − m)-number of y-indices with other indices unchanged.
For example, a 0 7 3 -brane (T 0 7 3 ∝ g −3 s ) associated with the winding coordinate y a 1 ···a 6 y,b 1 ···b 6 y that includes two y is mapped to a 1 6 3 -brane with the winding coordinate y 1 a 1 ···a 6 y,b 1 ···b 6 . According to the above dictionary, the whole M-theory coordinates (x I ) are mapped to the type IIB coordinates, where m, n, p = 1, . . . , d − 1 and α, β = 1, 2 . In [53], the map between the M-theory coordinates and the type IIB coordinates was expressed as and by using the same matrix S I M , the generalized metric was also transformed as Then, with the help of Buscher-like transformation rules for supergravity fields, the generalized metric M MN is nicely parameterized with the type IIB supergravity fields (see [53] for the details). In the following, we use the parameterization of M MN and obtain the brane action for a (p, q)-string. More detailed discussions and actions for other type IIB branes will be reported elsewhere.

Action for a (p, q)-string
When we considered M-branes we chose the M-theory section (3.10), but here we choose the type IIB section, where the last row in E M has been abbreviated for simplicity and we have defined (4.14) Furthermore, similar to (3.22), we introduce the scalar field e ω(σ) into the generalized metric M MN instead of the overall factor |G| 1 9−d (see Eqs. (2.12)-(2.17) in [53]). In the case of a string, which corresponds to the η-symbol η α MN (see [53]), the η-form becomes where (q α ) ≡ (p, q) are constants. Then, we consider a string action We can eliminate the auxiliary fields, P α m , P m 1 m 2 m 3 , P α m 1 ···m 5 , and P m 1 ···m 6 , m , by using their equations of motion, and the action becomes where we defined As in the case of the usual string action, the equation of motion for γ ab gives 20) and by using this, the action for X m and A α 1 becomes where e ω ≡ e ω |µ 1 | |q| . The equations of motion for ω show that e ω = |µ 1 | |q|, and we finally obtain This is the well-known (p, q)-string action [71,72] for (q α ) = (p, q). We can also show that the self-duality relation is satisfied (4.23)

Exotic branes and gauge fields in the external space
In the previous sections, we have considered only the internal components of the supergravity fields, such as C i 1 i 2 i 3 and C i 1 ···i 6 . Here, we also consider the external components, such as C µi 1 i 2 and C µ 1 µ 2 i 1 ···i 4 , where µ runs over the external (11 − d) dimensions. In fact, the external n-form gauge fields make up the so-called R n -representation of the E d(d) group (see [73]). We denote these external fields as where the index I n (n = 1, 2, 3, . . .) transforms in the R n -representation of the E d(d) (note that I 1 = I and I 2 = I in our M-theory parameterization).
Recently, while this manuscript was being prepared, [74] appeared on arXiv that constructed a U -duality-covariant action for strings, including the external fields as well. In our convention, their action takes the form where DY I ≡ dY I − A I + A I and A I ≡ A I µ dX µ . If the external fields (i.e. g µν , A I , and B µν; I 2 ) are ignored, their action reproduces our 1-brane action (4.16) by identifying dY I with our E I . As a natural extension, it is important to introduce external fields {A I 1 µ , B µ 1 µ 2 ; I 2 , . . .} up to the (p + 1)-form into our p-brane actions. If all of the external fields are introduced in a gauge-invariant manner, it will be possible to reproduce the actions for a KKM and exotic branes as we discuss below.
In order to argue that the extension of our p-brane action can completely reproduce the Wess-Zumino couplings for exotic branes, let us review which potentials are included in the . Only the supergravity fields that couple to branes with co-dimension higher than one are explicitly shown. external fields. For simplicity, let us first consider branes with co-dimension higher than two.
In M-theory, this means 7-branes or lower-dimensional branes. They are standard objects in M-theory (i.e. M2, M5, and KKM) that couple to standard fields (i.e. C 3 , C 6 , and C 7,1 ). As one can see from Table 1, an external p-form contains only the standard fields when we consider the E d(d) group with d ≤ 8 − p. The external p-form field comes to contain non-standard supergravity fields when we consider the E d(d) group with d = 9 − p. The non-standard potentials, C 9,3 and C 9,6 , are known to couple to defect branes (i.e. co-dimension 2-branes) known as the exotic 5 3 -brane and 2 6 -brane, respectively.
In order to reproduce whole actions for a KKM in the E d(d) exceptional spacetime with 1 ≤ d ≤ 8, we need to include external fields up to the 8-form, These external fields include all components of Cμ 1 ···μ 9 , i 1 i 2 i 3 . Similarly, the exotic 2 6 -brane appears only for 6 ≤ d ≤ 8, and in order to write down the action, we may only need If our expectation is correct, the exotic 2 6 -brane will be the most tractable example. We may also consider co-dimension-1 branes and co-dimension-0 branes that couple to non-standard supergravity fields hidden in the ellipses in Table 1 (see [75] for a recent study on mixedsymmetry potentials and the associated co-dimension-1 branes). Further investigation along this direction will be interesting.

Conclusion
In this paper, we showed that the action of the form can reproduce the conventional M-brane actions in a uniform manner. In the case of the M5-brane, the intrinsic metric γ ab naturally reproduced the 5-brane metric as a result of the equations of motion, and by using this metric, the self-duality relation, was realized. In contrast to the conventional formulations of extended sigma models (i.e. double/exceptional sigma model), the worldvolume gauge fields, such as A 2 and A 5 , are naturally introduced inside E I , which essentially plays the role of dX I in the conventional formulations.
In order to show the applicability of our formalism to type IIB branes, we demonstrated that the well-known (p, q)-string action can be correctly reproduced. An extension of our p-brane action which includes external fields and actions for exotic branes was discussed in Sect. 5.
It will be interesting future work to reproduce all of the known brane actions in M-theory and type IIB theory. So far, actions of exotic branes are constructed only for the exotic 5 2 2 -branes and 5 2 3 -branes in type II theory [16,76,77] and the 5 3 -brane in M-theory [78]. By considering the E 9(9) exceptional spacetime or including external fields, it will be possible to reproduce the actions for these branes as well as the other exotic branes discussed in Sect. 5. Extended sigma models play an important role in describing string/brane dynamics in "stringy" backgrounds, such as non-Riemannian backgrounds (see [79] for the detailed analysis) and backgrounds with non-geometric fluxes called U -folds. It will be interesting to study concrete applications. we can trivially integrate the wrapped M5-brane action over the 3-torus, and the M5-brane action will become an effective 2-brane action. Then, a natural expectation (at least if we ignore the gauge field A 2 for simplicity) is that the wrapped M5-brane action will take the form of the 2-brane action (3.19) with the following η-form: In fact, this kind of charge appears if we consider a duality ration of the M2 charge Q I (M2) , where q i 1 i 2 i 3 is proportional to k (3) and (R i 1 i 2 i 3 ) I J is an E d(d) generator in the R 2representation (see Appendix A.2). This 2-brane with the charge Q I (M2') may be interpreted as a bound state of an M2-brane and wrapped M5-branes like the (p, q)-string. It will be interesting to perform a more detailed analysis and clarify its relation to the (p, q)-membrane discussed in [38].

A Conventions
A.1 Differential forms We employ the following conventions for differential forms on a worldvolume: (A.1)

A.2 E d(d) algebra and the R n -representation
In the M-theory parameterization, we decompose the E d(d) (d ≤ 7) generators as follows: Their commutation relations are given as follows [22]: R i 1 ···i 6 , R j 1 ···j 6 = − 6! · 6! 5! δ where D ≡ i K i i . We denote the representation of the E d(d) group that is composed of the external n-form fields as the R n -representation, whose dimensions are determined as follows [73]: The R 9−d -representation is always the adjoint representation and there is a symmetry in the dimensions, dim R n = dim R 9−d−n . In the M-theory parameterization, we decompose the index I n of the R n -representation as where the ellipses are not necessary when we consider the E d(d) group with d ≥ 9 − n. We may simply denote I 1 and I 2 as I and I, respectively.
The matrix representations of the E d(d) generators in the R 1 -representation are given as follows [22]:

A.3 η-symbols in the M-theory parameterization
The η-symbols η I = (η IJ; I ) and η I = (η IJ; I ) (η IJ; I = η JI; I and η IJ; I = η JI; I ) are constant matrices that connect the symmetric product of two R 1 -representations and the R 2representation. When we consider M-theory, we decompose the R 2 -representation as The two types of η-symbols, η I and η I , are simply related as η IJ; I = η IJ; I , (A. 19) as matrices. Their explicit matrix forms are determined in [50] and are given as follows (see [50] for the explicit form of η I ): The relation between the η-symbols (and the Ω-tensor) and the Y -tensor known in the literature has been shown in detail in [50] (see, in particular, Appendix B therein). Similar expressions for the η-symbols and the Ω-tensor that are suitable for type IIB theory are given in [50].