Connecting M-theory and type IIB parameterizations in Exceptional Field Theory

In the exceptional field theory, there are two natural parameterizations for the generalized metric; in terms of bosonic fields in the eleven-dimensional supergravity (M-theory parameterization) and the type IIB supergravity (type IIB parameterization). In order to translate various results known in the M-theory to the type IIB theory or vice versa, an explicit map between the two parameterizations will be useful. In this note, we present such a linear map. Comparing the two parameterizations under the linear map, we reproduce the known T-duality transformation rules for the supergravity fields. We also obtain the T-duality rules for non-geometric $\beta$-/$\gamma$-fields appearing in the type II theory on a torus.


Introduction
The Exceptional Field Theory (EFT) [1][2][3][4][5][6][7][8][9][10][11][12][13] has been developed for a manifestly U -duality covariant formulation of supergravities. The generalized metric M IJ (x), which describes the geometry of the extended spacetime, is one of the most important fundamental fields in EFT. By providing a parameterization of M IJ (x) in terms of bosonic fields in the elevendimensional supergravity (M-theory parameterization) we can reproduce the action of the eleven-dimensional supergravity from the EFT action. On the other hand, if we consider a parameterization in terms of the bosonic fields in the type IIB supergravity (type IIB parameterization) the action of the type IIB supergravity can also be reproduced [9,14]. In this sense, EFT successfully unifies the M-theory and the type IIB theory.
The M-theory parameterization [7] takes the following structure: whereM is a block diagonal matrix consisting of the metric G ij , while L 3 and L 6 are lower block-triangular matrices consisting of the 3-form potential A 3 and the 6-form potential A 6 , respectively. On the other hand, the type IIB parameterization [15,16] takes the form, Here,M is a block diagonal matrix consisting of the Einstein-frame metric G mn and scalar fields (ϕ, C 0 ), while L 2 , L 4 , and L 6 are lower block-triangular matrices consisting of the 2-form potentials (B 2 , C 2 ), the 4-form potential D 4 , and the 6-form potentials (B 6 , C 6 ), respectively.
In [17], we proposed a worldvolume action for a p-brane, and the background fields are introduced only through the combination M IJ . In order to compare our action with the actions for known branes, such as the M2-brane or the M5-brane, the explicit parameterizations for M IJ were important. Although only the M-theory branes are considered in [17], according to the duality between the M-theory and the type IIB theory, we expect our brane action can also reproduce actions for the type IIB branes if we adopt the type IIB parameterization for the generalized metric. In order to realize this expectation, it is convenient to find a linear map between the M-theory parameterization and the type IIB parameterization.
In this note, we find a constant linear map S = (S I M ) that satisfies with some identifications between the M-theory fields and the type IIB fields (see [3,18] and [19] for similar results connecting the type IIA and the type IIB theories). We show that, using the relation of fields in the eleven-dimensional supergravity and the type IIA supergravity, the above identifications are nothing but the T -duality transformation rules between the type IIA and the type IIB supergravity, known as Buscher's rules [20][21][22][23][24].
In both the M-theory and the type IIB theory, we can also consider a different parameterization for M IJ , called the non-geometric parameterization [16,[25][26][27][28][29][30][31][32], in terms of the non-geometric potentials introduced in [33,34]. Using the above linear map S, we find identifications of non-geometric potentials in the M-theory/type IIA supergravity and the type IIB supergravities. Namely, we obtain Buscher's rules for non-geometric potentials (see [35] for a recent work on a similar topic).

M-theory and type IIB parameterizations
In this section, we fix our conventions for the M-theory parameterization and the type IIB parameterization (see also Appendix A for our conventions).

M-theory parameterization
Following [2,7,16], we consider the following decomposition of the generalized coordinates (x I ) = (x i , y i 1 i 2 , y i 1 ···i 5 , y i 1 ···i 7 , i , · · · ) (i = 1, . . . , d) , (2.1) where x i are coordinates on the internal d-torus and the multiple indices are totally antisymmetrized and ellipses become relevant only for E d(d) with d ≥ 8 (see section 4 for comments on the E 8(8) case). According to this decomposition, an explicit parameterization of the generalized metric for d ≤ 7 is given by [7,16] M IJ = (L T 6 L T 3M L 3 L 6 ) IJ , (2.2)

4)
where n ≡ 11 − d is the dimension of the external space. Note that the sign of y i 1 ···i 5 is flipped compared to [16]. Note also that L 3 and L 6 have the form, In our convention, components of a generalized vector are normalized as (2.10)

Type IIB parameterization
In the type IIB case, we parameterize the generalized coordinates as [15,16,36,37] (x M ) = x m , y α m , y m 1 m 2 m 3 , y α m 1 ···m 5 , y m 1 ···m 6 , m , · · · (m = 1, . . . , d − 1 , α = 1, 2) , (2.11) where x m are coordinates on the internal (d−1)-torus and again the multiple indices are totally antisymmetrized and the ellipses become relevant only for The type IIB parameterization of the generalized metric was found in [15,16] from a group theoretical approach and the explicit form for d ≤ 7 is given by Note that the sign conventions for y m 1 m 2 m 3 , y m 1 ···m 6 , m , and B α mn are opposite compared to [16]. As we explain in section 3, we rederive the above parameterization by using the M-theory parameterization (2.2) and making suitable identifications of the generalized coordinates (2.1) and (2.11) together with suitable identifications of the background fields.
Here, we shall explain each field appearing in M MN (see [16] for more details). The metric G mn is (the internal part of) the Einstein-frame metric in the ten-dimensional type IIB supergravity. It is related to (the internal part of) the string-frame metric g mn by Using the string-frame metric,M becomeŝ The SL(2) matrix m αβ is given by The antisymmetric symbols ǫ αβ and ǫ αβ are defined by ǫ 12 = 1 = ǫ 12 , and the inverse metric is an SL(2) S-duality doublet and the 4-form potential D 4 is S-duality invariant. We can define another Ramond-Ramond 4-form C 4 as C 4 ≡ D 4 + 1 2 B 2 ∧ C 2 , or, in components, A pair of 6-form potentials, transforms as an S-duality doublet. We also define (C m 1 ···m 6 , B m 1 ···m 6 ) as [16] 1 (2.24) The SL(2) transformation rules are summarized as (2.25) In particular, the S-duality transformation corresponds to Λ = 0 −1 1 0 . From these we obtain the S-duality transformation rules, (2.26) 1 Note that the sign conventions of Bmn, Bm 1 ···m 6 , and Bm 1 ···m 6 are also different from [16].
A generalized vector in the type IIB parameterization is normalized as (2.27)

Linear map between M-theory/type IIB parameterizations
In this section, we construct an explicit linear map between the two parameterizations. In order to find a map, we decompose the physical coordinates on the M-theory side as where x z is supposed to be the coordinate on the M-theory circle and x y is an arbitrary coordinate among the remaining (d − 1) coordinates. On the other hand, on the type IIB side, we consider the following (d − 2) + 1 decomposition: We then find a map between the M-theory compactified on a 2-torus (with coordinates x α ) and the type IIB theory compactified on a circle (with a coordinate x y ). This map precisely corresponds to the duality proposed in [38,39] and the SL(2) symmetry in the type IIB theory (2.25) corresponds to the SL(2) symmetry in the M-theory which changes the modular parameter on the 2-torus.

Linear map
We consider the following GL(d − 2) × SL(2)-covariant linear map between generalized vectors in the two parameterizations (see Appendix B for the convention for the combinatoric factors):  In terms of the generalized coordinates, we consider the following mappings:  .
Note that some of the above identifications appear in (6.10) of [3] and (4.7) and (4.8) of [18], where the identifications are found from a different argument.
After some tedious calculation (see Appendix B for the details), one can show that the generalized metric M IJ given in (2.2) is mapped to M MN given in (2.12), namely, under the following identifications: where we defined As we show below, the identifications (3.8) are precisely Buscher's rules after rewriting the fields in the eleven-dimensional supergravity in terms of those in the type IIA supergravity.

Conventional Buscher's rules
The supergravity fields in the eleven-dimensional and the type IIA supergravity are related by where p, q, r = 0, . . . , 9, and recall that x z is the coordinate on the M-theory circle. Plugging these into (3.8), we obtain the following relations: The above transformation rules completely match the transformation rules (under a Tduality along x y direction) obtained in [24] (see Appendix A therein) if we make the following identifications for the supergravity fields: As usual, the transformation rule for the Ramond-Ramond potentials can be summarized as If we define Ramond-Ramond poly-forms as 20) which appear in the D-brane worldvolume action, the n-form parts of A and A, denoted by A n and A n , satisfy a simple T -duality transformation rule [40] (see Appendix A)
The familiar supergravity backgrounds can be described well by the former parameterization, but the latter non-geometric parameterization is necessary when we consider non-geometric backgrounds, such as T -folds or the non-Riemannian background studied in [41]. Therefore, both parameterizations are equally important. The non-geometric parameterization for the generalized metric in EFTs was considered in [42,43] for the SL(5) EFT and in [16] for E d(d) . Following the convention of [16], the non-geometric parameterization in the M-theory is given by On the other hand, the non-geometric parameterization in the type IIB theory is given by [16] Note that we can easily obtain the identifications of non-geometric potentials between the M-theory and the type IIB theory without repeating a similar calculation as that performed above. In fact, the parameterization (3.22) for M IJ can be obtained by calculating the inverse, 2) and making the replacements, Similarly, the parameterization (3.26) for M MN is also obtained by calculating the inverse, M MN ≡ (M −1 ) MN of (2.12) and making the replacements, (3.33) Then, using which follows from (3.7), and making the replacements (3.32) and (3.33), we obtain , Ω pqz = β pq , Ω pqr = γ pqr , we obtain the following Buscher's rules for non-geometric potentials: g ab =g ab −g aygby − β ay β bỹ g yy ,g ay = β aỹ g yy ,g yy = 1 g yy , β ab = β ab + β aygby −g ay β bỹ g yy , β ay =g aỹ g yy , e −2φ = e −2φ g yy , γ a 1 ···an = γ a 1 ···any + n γ [a 1 ···a n−1 β an]y + n (n − 1) γ [a 1 ···a n−2 |y| β a n−1 |y|gan]ỹ g yy , where the inverse of the string-frame metric is denoted byg mn ≡ (g −1 ) mn . If we define the wedge product for p-vectors, v = (1/p!) v i 1 ···ip ∂ i 1 ∨ · · · ∨ ∂ ip , in the same manner as p-forms, and define poly-vectors, their n-vector parts, α n and α n , obey a simple T -duality transformation rule, α a 1 ···an = α a 1 ···any , α a 1 ···a n−1 y = α a 1 ···a n−1 , similar to the modified Ramond-Ramond forms A n and A n . Note that the non-geometric P -fluxes defined in [16] can be expressed as where ∨ is defined to commute with the exterior derivative (see section 4.2 of [32] where the above expression has been conjectured).
A family of co-dimension 2 branes, called an exotic p 7−p 3 -brane (0 ≤ p ≤ 7) (see [44] for the notation), is known to be a magnetic source of the P -flux [31,32,[45][46][47][48][49][50]. In particular, the 7 3brane, which is also known as the NS7-brane, is the S-dual of the D7-brane and it magnetically couples to the γ 0 potential in the same way as the D7-brane magnetically couples to the C 0 potential. On the other hand, an object which electrically couples to the C 0 potential is known as the D-instanton or the D(−1)-brane [51], and similarly an object which electrically couples to the γ 0 potential will also be an instanton, to be called the D(−1)-brane, whose mass is proportional to the string coupling constant g s (bound states of these instantons are known as Q-instantons [52]). Under multiple T -duality transformations, the D(−1)-brane is mapped to another object. From the transformation rule (3.40), the potential γ 0 = α 0 is also mapped to an n-vector α a 1 ···an (or α a 1 ···an ), and the object can be identified as the electric source of α a 1 ···an . Since α a 1 ···an is a 0-form, such an object will also be an instanton, as discussed in [47,50]. A similar object which couples to the β-field has been studied recently in [50,[53][54][55], and it will be interesting to study these instantons in more detail.

Summary and Outlook
In this note, we made a connection between the M-theory parameterization and the type IIB parameterization for E d(d) EFT with d ≤ 7, and obtained the duality transformation rules for non-geometric potentials in EFT. Our linear map will be useful when we concretely apply various results in EFT to the M-theory or the type IIB theory.
We can generalize the same calculation to the E 8(8) EFT. In this case, the decomposition of the 248 generalized coordinates (suitable for the M-theory parameterization) is given by [13,18,56,57] (x I ) = ( x i P [8] , y i 1 i 2
In terms of the type IIB theory, the decomposition of the 248 generalized coordinates is given by [13,57,60] (x M ) =( x m P [7] , y α m
In the E 8(8) EFT, from the tensor structure, we can consider the identifications between the generalized coordinates given in Table 1. There, c 1 , . . . , c 6 are constants to be determined by requiring that the (d − 2)-dimensional tensors are reorganized into (d − 1)-dimensional tensors in the type IIB theory. A small calculation suggests c 1 = c 3 = c 4 = c 5 = c 6 = 1, and a complete analysis will be performed in future work.
The following are some applications of our results: • If we have a solution of the eleven-dimensional supergravity that depends on eleven coordinates x i , by substituting the background fields into the generalized metric M IJ (x i ), M IJ (x i ) satisfies the equations of motion of EFT. In principle, we can read off the type IIB fields from the same generalized metric M IJ (x i ), and at the same time, we can also rename the coordinates x i as (x a , y α y ) using the linear map (3.6). This is still a solution of EFT, but it has dual-coordinate dependence, much like the localized monopole solution discussed in [65,66]. Thus, it is not a solution of the conventional type IIB supergravity.
Further studies on such non-geometric solutions in EFT will be interesting.

(4.3)
As we can see from the parameterization of the generalized metric, y iz in the type IIA theory or y 1 m in the type IIB theory can be identified with the dual coordinatesx m in the double field theory. Therefore, following the idea of [67,68], the above brane may be interpreted as a Dp-brane. In EFT, we can proceed further since we also have the S-duality transformation. In the following, we consider various rotations of the same 9-brane in the exceptional spacetime; let us call the brane a (· · · )-brane if the brane is extending in the (· · · ) directions. Performing the S-duality transformation, a (x 0 , . . . , x 5 , y 1 6 , . . . , y 1 9 )-brane (D5-brane) is mapped to a (x 0 , . . . , x 5 , y 2 6 , . . . , y 2 9 )-brane, which will be interpreted as the type IIB NS5-brane. Further performing a T -duality along the x 6 direction, we obtain a (x 0 , . . . , x 5 , x z , y 67 , y 68 , y 69 )-brane, which may be the Kaluza-Klein monopole in the M-theory, where x 6 will be the Taub-NUT direction.
If we swap the M-theory circle x z and the Taub-NUT direction, we obtain a D6-brane, (x 0 , . . . , x 6 , y 7z , y 8z , y 9z ). A (x 0 , . . . , x 5 , y 1 6 , y 1 7 , y 678 , y 679 )-brane may be interpreted as the exotic 5 2 2 -brane. Exchanging x 2 and x z in the D2-brane, (x 0 , x 1 , x 2 , y 3z , . . . , y 9z ), we obtain a (x 0 , x 1 , x z , y 23 , . . . , y 29 )-brane, which may be the type IIA fundamental string. If this viewpoint works well, various known branes may be interpreted as a single object that is wrapping on various dual directions. This viewpoint is consistent with the proposal that strings and branes are waves propagating in various dual directions [66,69,70]. It will be interesting to develop this viewpoint further by finding a U -duality invariant action for the brane in the exceptional spacetime, and also by showing how various brane tensions can be reproduced. Acknowledgment Y.S. would like to thank Soo-Jong Rey and Kanghoon Lee for useful discussions and collaborations on related topics.

A Conventions
In this appendix, we summarize our conventions.

A.1 Conventions for supergravity fields
We clarify our conventions for the supergravity fields by displaying various expressions.
In the eleven-dimensional supergravity, the gauge transformations of the gauge potentials can be read off from the parameterization (2.2): We define the invariant field strengths, which satisfy the Bianchi identity, If we also consider diffeomorphisms, the gauge symmetries become Considering the dimensional reduction to the type IIA theory, the gauge potentials and the field strengths are decomposed as where we defined the field strengths, which satisfy the Bianchi identities From the gauge symmetries in the eleven-dimensional supergravity, we obtain the gauge symmetries in the type IIA supergravity, The gauge symmetries in the type IIB supergravity, with our convention (2.12), become (A.9) These are equivalent to (A.10) The field strengths in the type IIB supergravity are defined by and these satisfy the Bianchi identities, If we define poly-forms, we obtain (A.14) A convenient expression for the T -duality rule is as follows. If we consider a T -duality along the x y direction in a type IIB background, ds 2 = g ab dx a dx b + g yy dx y + a 1 2 , e 2ϕ , we obtain the following dual background on the type IIA side, Using this rule, we can show that modified Ramond-Ramond potentials A ≡ e −B 2 C and A ≡ e −B 2 C follow a simple transformation rule. Indeed, in the original type IIB background, we have 17) while A in the transformed background becomes Namely, the transformation rule is simply given by

A.2 Period of generalized coordinates
We choose the periods of the physical coordinates in the M-theory and the type IIB theory as where the radius of the M-theory circle is identified as R z = g s l s . The Planck length in eleven dimensions is given by s l s . From our linear map and the transformation rules, T -duality : R → l 2 s /R , g s → g s l s /R , l s → l s , S-duality : g s → 1/g s , l s = g 1/2 s l s , we can postulate the following radius in the dual directions: M-theory:

B Outline of the computation
Here, we explain a way to check the linear map (3.7) and (3.8).

B.1 M-theory: decomposition into (d − 2) + 2 dimensions
If we consider, for example, the combination, under the (d − 2) + 2 dimensions, it can be expanded as Following the rule, we decompose a generalized vector as and correspondingly decompose the generalized metric as follows:  As in the case of the M-theory, we decompose a generalized vector in the type IIB section as and also the generalized metric as

B.3 Similarity transformation
In order to examine the map (3.7), it is convenient to use (3.8) from the beginning, although we have found (3.8). Using (3.8), the metric and the inverse metric become where Λ −T j 1 ···jp . In terms of the type IIB fields, the nonvanishing components of G i 1 ···ip, j 1 ···jp and Λ −T j 1 ···jp i 1 ···ip , and also |G| are given by We can also show that     After laborious calculations, one can show that S T M S is equal to (B.12).