**1. Introduction**

The notion of symmetry has been one of the most important drivers for theoretical physics in recent decades. Recently, of most interest have been duality symmetries relating different theories or different regimes of the same theory. The most well-known example of the former is the renowned AdS/CFT correspondence of Maldacena [1], and more generally gauge-gravity duality. This correspondence relates string theory on *AdS*5 × S5 at small coupling and the quantum N = 4 SYM at strong coupling. This is based on equivalence between two descriptions of the same near-horizon region of the D3-brane in terms of the open and closed strings, and allows addressing phenomena in CFT at strong coupling in terms of gravitational degrees of freedom at weak coupling.

String theory also possesses symmetries, called T- and S-dualities, which relate different regimes of the same theory. These allowed unifying five string theories, Type IIA/B, Type I, heterotic *O*(32) and *E*8 × *E*8, into a duality web and to understand all these as different regimes of a single theory, usually addressed as M-theory [2,3]. T-duality is a perturbative symmetry of the fundamental string and is the oldest known duality in string theory. It manifests itself in the amazing fact that Type IIA and Type IIB string theories compactified on a 1-torus S1 are equivalent at quantum level. Most transparently this can be seen when looking at the mass spectrum of the closed string on a background with one circular direction of radius *R*

$$M^2 = p^2 + \frac{n^2}{R^2} + \frac{m^2 R^2}{a'^2} + 2(N + \bar{N} - 2). \tag{1}$$

Here *n* is the number of discrete momenta of the string along the circular direction and *m* is the number of windings of the string around the cycle, *N* and *N*¯ correspond to the numbers of higher level left and right modes on the string. One immediately notices that the mass spectrum is symmetric under the following replacement

$$\begin{array}{c} R \longleftarrow \frac{\alpha'}{R}, \\ m \longleftarrow n. \end{array} \tag{2}$$

The symmetry relates backgrounds with radii *R* and *a* /*R* upon switching string momentum and winding modes. More generally T-duality mixes metric and 2-form gauge field degrees of freedom, thus potentially completely messing up structure of space-time. In particular, the symmetry allows consistently defining string dynamics on such backgrounds, given by explicit metric *<sup>G</sup>μν* and the Kalb-Ramond field *<sup>B</sup>μν* that cannot be consistently described in terms of manifolds. Instead, these are related to as T-folds, which are defined as a set of patches glued by T-duality transformations, rather than diffeomorphisms [4]. Such backgrounds are called non-geometric and are of huge interest for cosmological model building as string vacua potentially capable of completely stabilizing scalar moduli ending up with a dS-like space with small cosmological constant.

T-duality is a perturbative symmetry of string theory seen already in the mass spectrum of string excitations. S-duality of Type IIB string theory provides an example of non-perturbative string symmetry. This relates strong and weak coupled regimes of the theory, and in addition relates heterotic SO(32) and Type I strings. The net of dualities between five string theories allows understanding them as different approximations to a single 11-dimensional theory called M-theory. The 11th direction arises in the strong coupling limit of Type IIA theory.

M-theory describes dynamics of M2 and M5 branes, which are fundamental 2 and 5 dimensional objects interacting with 3-form and 6-form gauge potentials dual to each other [3]. Low-energy limit of the theory is captured by 11-dimensional maximal supergravity, whose algebra of central charges is nicely interpreted in terms of M2 and M5 brane charges [3]. M-theory compactified on a circle S1 gives Type IIA theory with the fundamental string arising from wrapped M2 brane. Compactifying M-theory on a 2-torus one recovers either Type IIA or Type IIB depending on the reduction scheme, which is the fundamental precursor for T-duality symmetry between these theory. In addition, modular symmetry of the torus gives rise to S-duality symmetry of Type IIB as will be explained in more details below. Together, T- and S-duality of the string combine into a set of U-duality symmetries, which appears to be a powerful tool for investigating properties and the internal structure of M-theory. This review is focused on duality symmetries of string and M-theory and in particular at approaches to supergravity covariant with respect to these symmetries.

#### *1.1. Dualities in String Theory*

S-duality is a hidden symmetry of the Type IIB string theory which relates strong and weak coupling regimes. On the field theory level this can be recovered by inspecting spectrum of massless modes of the Type IIB string, which includes

*g*, *φ*, *<sup>B</sup>*(2), *<sup>C</sup>*(0), *<sup>C</sup>*(2), *<sup>C</sup>*(4), (3)

where *g* is the metric, *φ* is the scalar field called the dilaton and the 4-form gauge field *<sup>C</sup>*(4) is defined to have self-dual field strength. With the action of the S-duality group SL(2, R) the fields drop into irreducible representations with *g* being a scalar, the 2-forms *<sup>B</sup>*(2), *<sup>C</sup>*(2) combine into a doubled and the scalar fields *φ*, *<sup>C</sup>*(0) combine into the so-called axio-dilaton defined as

$$
\pi = \mathbb{C}\_{(0)} + i e^{-\Phi}.\tag{4}
$$

Axio-dilaton transforms non-linearly under S-duality

$$
\pi' = \frac{a\pi + b}{c\pi + d}, \quad \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \text{SL}(2, \mathbb{R}).\tag{5}
$$

The existence of this hidden symmetry of the Type IIB massless string spectrum suggests that the supergravity action can be rewritten in an SL(2, R)-covariant form, which is indeed possible and results in the following (see e.g., [5]):

$$\begin{split} S\_{IIB} &= -\frac{1}{2} \int d^{10}x \left( R - \frac{1}{4} Tr(\partial M \partial M^{-1}) + \frac{3}{4} H\_{\mu\nu\rho}{}^{I} M\_{I\{}} H^{\mu\nu\rho\}} \\ &+ \frac{5}{6} F\_{\mu\nu\rho\sigma} F^{\mu\nu\rho\sigma} + \frac{1}{96\sqrt{-\mathcal{S}}} \varepsilon\_{I\{}} \mathbb{C}\_{(4)} \wedge H\_{(3)}{}^{I} \wedge H\_{(3)}{}^{J} \right), \end{split} \tag{6}$$

where the indices *I*, *J*, ··· = 1, 2 label the fundamental **2** representation of SL(2) and

$$\begin{aligned} M\_{I\!\!\!I} &= \frac{1}{\heartsuit\pi} \begin{bmatrix} |\pi|^2 & -\Re\pi \\ -\Re\pi & 1 \end{bmatrix}, \\\ H\_{\mu\nu\rho}{}^I &= \partial\_{[\mu}B\_{\nu\rho]}{}^I, \quad F\_{\mu\nu\rho\sigma\kappa} = \partial\_{[\mu}\mathcal{L}\_{\nu\rho\sigma\kappa]} + \frac{3}{4}\epsilon\_{I\!\!f}B\_{[\mu\nu}{}^I\partial\_\rho B\_{\sigma\kappa]}{}^I. \end{aligned} \tag{7}$$

One may notice that the transformation (5) has the form that of the transformation of complex structure of a 2-torus. This observation leads to the geometrical picture which is behind F-theory, a 12-dimensional field theory, whose compactification on a 2-torus gives an orientifold reduction of Type IIB theory (for more details see e.g., [6,7]). Freedom in definition of a complex structure on the 2-torus of F-theory is equivalent to the S-duality symmetry of the 10-dimensional theory. Such geometric interpretation of a symmetry of a theory goes along the line of the old Kaluza-Klein idea, where the U(1) gauge symmetry of Maxwell theory is lifted into reparametrisations of a small circle (1-torus) of a 5-dimensional gravitational theory with no electromagnetic degrees of freedom. In this short review we highlight basic features and list some applications of the so-called Doubled (Exceptional) Field Theory, which does the same to T(U)-dualities of string (M-)theory, i.e., provides a geometric interpretation of the duality symmetries in terms of geometry of an especially constructed higher dimensional space.

T-duality is a hidden symmetry of the 2-dimensional non-linear sigma model (string theory) which relates the theory on a torus with radius *Rx* of a given direction *x* and the same theory on a torus with radius *α* /*Rx* of the same direction. Under T-duality transformation of the string background, given by the metric, Kalb-Ramond 2-form field, the dilaton and the RR fields, partition function of the string does not change. Consider the action for the closed string on a background with one circular direction in the conformal gauge and adopt the light cone world-sheet coordinates *σ*±

$$\begin{split} \mathcal{S}\_{1}[\theta] &= \int d^{2}\sigma \left( G + B \right)\_{\mu\nu} \partial\_{+} X^{\mu} \partial\_{-} X^{\nu} \\ &= \int d^{2}\sigma \left( G\_{\theta\theta} \partial\_{+}\theta \partial\_{-}\theta + E\_{\theta\theta} \partial\_{+} X^{\mu} \partial\_{-}\theta + E\_{\theta\theta} \partial\_{+} \theta \partial\_{-} X^{\mu} + E\_{\theta\beta} \partial\_{+} X^{\mu} \partial\_{-} X^{\beta} \right). \end{split} \tag{8}$$

Here *μ*, *ν* = 0, ... , 9 label all ten space-time directions, *θ* = *<sup>θ</sup>*(*<sup>σ</sup>*±) parametrizes the compact direction and *α*ˆ, *β*ˆ = 0, ... , 8 parametrize the rest. For the background fields we define *E* = *G* + *B*. To see that the action above is invariant under replacing the circle S1 *θ* of radius *R* by a circle S1 *λ* of the inverse radius 1/*R* parametrized by *λ*, one used the global symmetry *θ* → *θ* + *ξ* and turns it into a local one. The gauging is performed by introducing a world-volume 1-form gauge field *A* = *<sup>A</sup>*+*dσ*<sup>+</sup> + *A*−*dσ*<sup>−</sup> and replacing normal derivatives by covariant e *dθ* → *Dθ* = *dθ* + *A*. To turn back to the correct counting of the degrees of freedom in the theory one must introduce a Lagrange multiplier to restrict the gauge field to pure gauge. Hence, one arrives at the following action

$$\mathcal{S}\_2[\theta, \lambda\_\prime A] = \mathcal{S}\_1[d\theta \to D\theta] + \int \lambda F\_\prime \tag{9}$$

where *F* = *dA* is the field strength for the gauge field *A*. Integrating the Lagrange multiplier *λ* out of the partition function one return back to the action *<sup>S</sup>*1[*θ*]. Alternatively, integrating out vector degrees of freedom *A* one arrives at (for more details see [8])

$$S\_2[\lambda] = \int d^2 \sigma \left( G\_{\lambda \dot{\lambda}}^{\prime} \partial\_+ \lambda \partial\_- \lambda + E\_{\hbar \dot{\lambda}}^{\prime} \partial\_+ X^{\underline{\hbar}} \partial\_- \lambda + E\_{\lambda \underline{\hbar}}^{\prime} \partial\_+ \lambda \partial\_- X^{\underline{\hbar}} + E\_{\hbar \underline{\hbar}}^{\prime} \partial\_+ X^{\underline{\hbar}} \partial\_- X^{\underline{\hbar}} \right), \tag{10}$$

with the new background *E* = *G* + *B* defined by the so-called Buscher rules

$$\begin{aligned} G'\_{\lambda\lambda} &= \frac{1}{G\_{\theta\theta}},\\ E'\_{\lambda\dot{\theta}} &= \frac{1}{G\_{\theta\theta}} E\_{\theta\dot{\theta}},\\ E'\_{\dot{\lambda}\dot{\lambda}} &= -\frac{1}{G\_{\theta\theta}} E\_{\lambda\theta},\\ E'\_{\dot{\lambda}\dot{\beta}} &= E\_{\theta\dot{\beta}} - E\_{\theta\theta} \frac{1}{G\_{\theta\theta}} E\_{\theta\dot{\beta}}.\end{aligned} \tag{11}$$

Since the partition function did not change during the above procedure, the physics of the string on two background related by the T-duality transformation (11) is the same. Taking into account the transformation of measure of the functional integral one supplements the above rules by the following transformation of the dilaton:

$$
\varphi' - \frac{1}{4} \ln \det G' = \varphi - \frac{1}{4} \ln \det G.\tag{12}
$$

In the more general case of the string on a background with *d* compact isometric directions the group of T-duality transformations can be shown to be <sup>O</sup>(*d*, *d*;<sup>Z</sup>). The most convenient tool for that is the so-called Duff's procedure [9,10] which exploits hidden symmetry between equations of motion and Bianchi identities following from the action for the non-linear sigma model

$$S = \int d^2 \sigma \left( \sqrt{-h} \, h^{ab} G\_{\mu \nu} + \epsilon^{ab} B\_{\mu \nu} \right) \partial\_a X^\mu \partial\_b X^\nu \tag{13}$$

For the metric and the *B*-field equations of motion and Bianchi identities can be rewritten in explicitly <sup>O</sup>(*d*, *d*;<sup>Z</sup>)-covariant form

$$
\hbar \eta\_{MN} \Phi^{iN} = \mathcal{H}\_{MN} \Phi^{iN}.\tag{14}
$$

Here we define combinations

$$\Phi^{iM} = \begin{bmatrix} \epsilon^{ab}\partial\_b X^\mu\\ \epsilon^{ab}\partial\_b Y\_\mu \end{bmatrix}, \quad \Phi^{iM} = \begin{bmatrix} \sqrt{-h} \, h^{ab} \partial\_b X^\mu\\ \sqrt{-h} \, h^{ab} \partial\_b Y\_\mu \end{bmatrix} \tag{15}$$

of derivatives of the normal coordinates *Xμ* and the dual coordinates *<sup>Y</sup>μ*. The equation (14) can be considered to be self-duality constraints, which remove half of the fields of the full doubled set X *M* = ( *<sup>X</sup>μ*,*Yμ*). The matrix H*MN* parametrizes the background fields in a T-duality covariant manner

$$\mathcal{H}\_{\rm MN} = \begin{bmatrix} G\_{\mu\nu} - B\_{\mu\rho} B^{\rho}{}\_{\nu} & -B\_{\mu}{}^{\nu} \\ B^{\mu}{}\_{\nu} & G^{\mu\nu} \end{bmatrix} \in \frac{O(d,d)}{O(d) \times O(d)}.\tag{16}$$

The invariant tensor of the <sup>O</sup>(*d*, *d*) group *ηMN* is taken in the block-skew-diagonal form

$$
\eta\_{MN} = \begin{bmatrix} 0 & \delta^\mu\_\nu \\ \delta^\mu\_\nu & 0 \end{bmatrix}. \tag{17}
$$

At the level of string theory T-duality is a proper symmetry of the theory, which does not change physics upon a transformation. When reducing to the low-energy dynamics governed by 10-dimensional half-maximal supergravity, T-duality turns into a solution-generating symmetry, as it transforms a given string theory background into another one. In this case, the symmetry group

<sup>O</sup>(*d*, *d*; R)is defined over rational numbers rather than only integers. In what follows, we will always denote this group simply by <sup>O</sup>(*d*, *d*), and add Z explicitly when needed.

#### *1.2. U-duality in Maximal Supergravity*

When combined into the web of dualities five string theories become a single 11-dimensional M-theory, encoded in dynamics of M2 and M5 branes. T- and S-duality symmetries lift into U-dualities of the membranes; however, these are much more complicated for a sigma model analysis. In the seminal paper by Cremmer, Julia, Lu and Pope [11,12] it has been shown that 11-dimensional supergravity compactified on a *d*-torus T*<sup>d</sup>* possesses hidden symmetry *Ed*(*d*). This is reflected in the fact that all bosonic field of the reduced theory can be collected into irreducible representations of the U-duality group, while fermionic fields transform under maximal compact subgroup of *Ed*(*d*). Here the notation *d*(*d*) means that one takes maximal real subgroup of complexification of the group *Ed*.

The most transparent way to see the symmetry is to analyse spectrum of fields in the lower dimensional theory obtained by reduction of the 11-dimensional fields *<sup>G</sup>μ*<sup>ˆ</sup>*ν*ˆ, *<sup>C</sup>μ*<sup>ˆ</sup>*ν*<sup>ˆ</sup>*ρ*<sup>ˆ</sup> say to 4 dimensions.

As it is shown in Figure 1 the resulting fields can be collected into the vector fields <sup>A</sup>*μ<sup>M</sup>* transforming in the **56** of *E*7, scalar coset *MMN* ∈ *<sup>E</sup>*7(7)/*SU*(8), *E*7 scalars *<sup>G</sup>μν* and a constant *<sup>C</sup>μνρ*. Vector degrees of freedom (*<sup>A</sup>μ m*, *<sup>A</sup>μmn*) coming from the metric and the 3-form field correspond 28 electric gauge potentials. To compose the **56** irrep of *E*7, which is representation of the lowest dimension, one adds magnetic potentials (*A*˜*μ<sup>m</sup>*, *<sup>A</sup>*˜*μmn*) and imposes self-duality condition on the U-duality covariant field strength

$$\mathcal{F}\_{\mu\nu}{}^{M} = \frac{i}{2} \epsilon\_{\mu\nu\rho\sigma} \Omega^{MN} M\_{NK} \mathcal{F}^{\rho\sigma K}. \tag{18}$$

Here Ω*MN* is the symmetric invariant tensor of *E*7 and *MMN* is the scalar matrix which parametrizes the coset space *<sup>E</sup>*7(7)/*SU*(8) and the antisymmetric tensor is usually chosen to be 0123 = *i*. The field strength is defined as usual as <sup>F</sup>*μν M* = <sup>2</sup>*∂*[*μ*A*ν*] *M*. As it has been explicitly shown in detail in [11], to end up with irreducible representations of the U-duality group one must dualize all tensor fields to the lowest possible rank. For the 2-form field *<sup>B</sup>μνm* one constructs gauge invariant 3-form field strength, whose Bianchi identities and equations of motion can be swapped by Hodge dualization to a 1-form field strength. This is associated with scalar fields. To keep covariancy and to recover tensor hierarchy one must add the same amount of 2-form fields and impose duality condition.

**Figure 1.** Reduction of 11D fields into four dimensions, dualization into the lowest possible rank tensor and recollection into the *E*7 multiplet in the **56** and the coset space *<sup>E</sup>*7(7)/*SU*(8). Both magnetic and electric vectors potentials are included in the counting (see the text).

Finally, equations of motion for the 3-form field in 4 dimensions imply that the field strength is constant. It appears that this constant is not a scalar under the U-duality group, and one must either set it to zero or to turn on all constant belonging to its representation. This can be done

consistently in embedding tensor formalism, which results in a deformed theory with non-abelian gauge symmetry [13].

In conventional supergravity U-duality symmetry is a global symmetry of the theory similar to the U(1) duality symmetry of equations of motion of electrodynamics. In exceptional field theories this symmetry receives a geometric interpretation as a symmetry of the underlying space-time. To carry a proper representation of a U-duality group *Ed* space-time coordinates must be extended following a special rule based on winding of branes, both standard and exotic. In Section 2.1 the construction and local symmetries of the extended space will be explained in more details, in Section 2.3 field spectrum and action of exceptional field theories will be reviewed for a general U-duality group and for *<sup>E</sup>*7(7) as an explicit example. We will show that this theory reproduces the full 11D supergravity, Type IIA/B, D = 4 gauged and ungauged supergravities depending on the choice of solution of a special constraint called section condition. In Section 3.1 non-conventional solutions of this condition will be shown to lead to non-geometric backgrounds, i.e., such field configurations which are not globally or even locally well defined in terms of differential geometry. Finally, in Section 3.2 we consider exotic branes as T(U)-duality partners of the standard branes of string and M-theory, and review their description in terms of extended space and non-geometry in exceptional field theories.

#### **2. Duality-Covariant Field Theories**

#### *2.1. Local Symmetries of Extended Space*

Duality symmetries appear in toroidal reductions of supergravity and combine geometric symmetries of the torus (diffeomorphisms), gauge transformations and actual duality transformations mixing space-time and gauge degrees of freedom. To proceed with construction of a duality covariant theory, one understands the group of duality symmetries *<sup>O</sup>*(*d*, *d*) or *Ed*(*d*) more fundamentally as descending from geometrical structure of the underlying space. Since *d* coordinates of the *d*-torus do not fit into an irreducible representation of the duality group, one has to consider an extended space.

In the previous section, we saw that T-duality symmetries relate winding and momentum modes of the string, and for the string on a torus T*<sup>d</sup>* the mass spectrum can be covariantly written as

$$M^2 = p^2 + \mathcal{H}\_{MN}\mathcal{P}^M\mathcal{P}^N + (N+\bar{N}-2),\tag{19}$$

where H*MN* is the generalized metric and P *M* combines the momentum *pm* and winding *w<sup>m</sup>* of the string in an <sup>O</sup>(*d*, *d*)-covariant vector

$$\mathcal{P}^M = \begin{bmatrix} p\_m \\ w^m \end{bmatrix}. \tag{20}$$

The first term *p*2 contains momenta in "external" non-toroidal directions *p*2 = *ημν pμ p<sup>ν</sup>*. Recalling Duff's procedure, that operates with normal *Xm* and dual *Ym* scalar fields on the world-volume of the string, it is natural to consider backgrounds, not necessarily toroidal that depend on the full set of T-duality covariant coordinates X *M* = (*xm*, *<sup>x</sup>*˜*m*). Note that the generalized metric in the mass formula is constant since the background is toroidal, this is relaxed in exceptional field theory and the toroidal case is understood as the most symmetric background preserving all duality symmetries.

At the level of the non-linear sigma model extra degrees of freedom encoded by the dual scalar fields *Ym* are removed by the self-duality condition (14). Similarly, T-covariant field theory with fields depending on the doubled amount of coordinates must be augmented by a constraint that reduces the number of space-time direction. Apart from reference to the sigma model, this is necessary for further supersymmetric completion of the theory to avoid higher spin fields, which normally appear in supersymmetric theories in dimensions higher than 11. The condition naturally follows

from construction of local diffeomorphism symmetry on the extended space parametrized by the coordinates X*<sup>M</sup>* which is defined by the so-called generalized Lie derivative

$$
\mathcal{L}\_{\Lambda}V^{M} = \Lambda^{N}\partial\_{N}V^{M} - \Lambda^{M}\partial\_{M}V^{N} + \mathcal{Y}\_{KL}^{MN}\partial\_{N}\Lambda^{K}V^{L},\tag{21}
$$

where *V<sup>M</sup>* = (*vm*, *<sup>ω</sup>m*) is a generalized vector combining a GL(*d*) vector *v<sup>n</sup>* and a 1-form *ωm*, the same for the transformation parameter Λ*<sup>M</sup>* = (*λ<sup>m</sup>*, *λ*˜ *<sup>m</sup>*). The tensor *YMN KL* encoding deformation of the generalized Lie derivative away from the conventional GL(*d*) Lie derivative is subject to constraints following from consistency of algebra of transformation *<sup>δ</sup>*Λ*V<sup>M</sup>* = <sup>L</sup>Λ*VM*. These constraints have been analysed in [14] and can be summarized as

$$\begin{split} & Y\_{KL}^{(MN}Y\_{PQ}^{R)L} - Y\_{PQ}^{(MN}\delta\_{K}^{R)} = 0 \text{, for } d \le 5, \\ & Y\_{KL}^{MN} = -a\_d \, P\_{K} \, {}^{M}{}\_{L}{}^{N} + \beta\_d \, \delta\_{K}^{M} \delta\_{L}^{N} + \delta\_{L}^{M} \delta\_{K}^{N} \\ & Y\_{KB}^{M4} \, Y\_{AL}^{RN} = (2 - a\_d) \, Y\_{KL}^{MN} + (D\beta\_d + a\_d) \, \beta\_d \, \delta\_{K}^{M} \delta\_{L}^{N} + (a\_d - 1) \, \delta\_{L}^{M} \delta\_{K}^{N} . \end{split} \tag{22}$$

Here *d* is the number of compact dimensions and *PABCD* is the projector on the adjoint representation of the corresponding duality group. It is defined as *PABCDPDCKL* = *PABKL* and *PABBA* = dim(adj). The coefficients *αd* and *βd* depend on the duality group and for the cases in question take numerical values (*<sup>α</sup>*4, *β*4)=(3, 15 ), (*<sup>α</sup>*5, *β*5)=(4, 14 ), (*<sup>α</sup>*6, *β*6)=(6, 13 ). These conditions imply that the Y-tensor must be constructed from invariant tensors of the corresponding T- and U-duality groups (see Table 1).

**Table 1.** *Y*-tensor for different T(U)-duality groups for string and M-theory on a *d*-torus. Here the Greek indices *α*, *β*, *γ* = 1, ... , 5 label the representation **5** of *SL*(5) and the index *i* labels the **10** of *SO*(5, <sup>5</sup>), *n* denotes dimension of the representation generalized vectors of the theory transform in.


The above still does not guarantee the algebra of transformations *<sup>δ</sup>*Λ*V<sup>M</sup>* is closed. Indeed, one writes

$$\begin{aligned} \mathcal{L}\_{\Lambda\_1} \mathcal{L}\_{\Lambda\_2} V^M - \mathcal{L}\_{\Lambda\_2} \mathcal{L}\_{\Lambda\_1} V^M &= \mathcal{L}\_{[\Lambda\_1, \Lambda\_2]\_\mathbb{C}} V^M + F\_0^M \\ F\_0^M &= \end{aligned} \tag{23}$$

where the bracket [<sup>Λ</sup>1, <sup>Λ</sup>2]*C* = L*<sup>L</sup>*1Λ<sup>2</sup> − L*<sup>L</sup>*2Λ<sup>1</sup> is a generalisation of the Courant bracket of the Hitchin's generalised geometry. The obstruction *<sup>F</sup>M*0 =for the algebra to close is proportional to terms of the type *<sup>η</sup>MN∂M*Φ1*∂N*Φ2, hence one naturally imposes the so-called section constraint

$$
\eta^{MN}\partial\_M\bullet\partial\_N\bullet = 0\tag{24}$$

where bullets stand for any combination of any fields. Similarly, one shows that the very same condition ensures satisfaction of the Jacobi identity for *δ*Λ.

The most natural and transparent solution of the section constraint is ˜ *∂<sup>m</sup>*• = 0, which is simply the condition that nothing depends on *<sup>x</sup>*˜*m*. This drops the generalized Lie derivative back to the conventional undoubled space-time and splits it into the usual Lie derivative and gauge

transformations. More generally one can solve the section constraint by dropping dependence on all *<sup>x</sup>*˜*m* apart a given *<sup>x</sup>*˜*d*, and drop in addition dependence on the corresponding normal coordinate *<sup>x</sup>d*. Now one can use the set {*x*1, ... , *xd*−1, *<sup>x</sup>*˜*d*} to measure distances, and hence these will correspond to geometric coordinates of the new frame. These two frames are related by a T-duality transformation along the direction *d*

$$T\_d: \mathfrak{x}^d \longleftrightarrow \mathfrak{x}\_d. \tag{25}$$

In the next subsection we construct Exceptional Field Theories, which do not distinguish between such frames, hence providing a local T(U)-duality covariant approach to supergravity.

#### *2.2. Winding Modes and Exotic Branes*

Before proceeding with the field theory construction it is suggestive to follow the logic of counting of winding modes of M-branes in M-theory in details. In contrast to the string, where the winding mode is always parametrized by a 1-form *ωm* irrespective of the number of compact directions, winding modes of branes follows more complicated pattern. To start with, one notices that winding modes of a *p*-brane can be parametrized by a *p*-form. Spectrum of M-theory contains M2, M5-branes, KK6-monopole and various additional (exotic) branes, whose counting will be useful for U-duality groups *<sup>E</sup>*8(8) and larger. Table 2 lists irreducible representations of U-duality groups for each dimension *d* of compact torus, governing transformation of extended coordinates X *M* and the corresponding generalized momentum P *M*. The normal geometric coordinates correspond to the usual momentum *P* of a state. Windings of M2 and M5 branes are given by 2- and 5-forms respectively and give *C*<sup>2</sup> *d* and *C*<sup>5</sup> *d* number of winding states, where *Cn m* is the binomial coefficient. Hence, the M5 brane contributes only starting from dimension *d* = 5, since it simply cannot wrap spaces of lower dimensions.

**Table 2.** Counting of winding modes of branes of M-theory on a background of the form *M*4 × T*<sup>d</sup>* with *M*4 being a four-dimensional manifold. The first column contains dimensions of the compact torus, the second column lists the corresponding U-duality group, and the last column lists representations of *G* under which coordinates of the extended space transform.


The Kaluza-Klein monopole *KK*6 is an object with 6 + 1-dimensional worldvolume and one Taub-NUT direction corresponding to the Hopf cycle. This is magnetic dual of the graviton. Hence, it windings are represented by a mixed-symmetry tensor *<sup>z</sup>*(7,1), which is a 7-form taking values in 1-forms and traceless. In components this is represented by the following tensor

$$
\mathbb{Z}\_{a\_1\dots a\_{\mathcal{T}}, b\_{\mathcal{T}}} \tag{26}
$$

where *b* must be equal to one of *ai*'s for non-vanishing components. For *d* = 7 winding direction this amounts in 7 independent winding modes, while for the *<sup>E</sup>*8(8) case is is suggestive to contract *za*1...*a*7,*<sup>b</sup>* with Levi-Civita tensor as

$$z\_b{}^a = z\_{a\_1\dots a\_7, b} \epsilon^{a\_1\dots a\_7 a}.\tag{27}$$

In total *za b* has 8 × 8 = 64 components and the condition *za a* no sum = 0 removes 8 more leaving only 56.

The important observation here is that the total number of momentum and winding modes for a *d*-torus with *d* < 8 sums up to dimension of an irreducible representation of the corresponding U-duality symmetry group. For *d* = 8 this apparently does not work, as summing winding modes of all branes up to the KK6 one obtains 148, while dimension of the smallest irrep is 248. To cure the result one first recalls that the spectrum of both string and M-theory contains exotic branes in addition to the normal (standard) branes. These are T(U)-duality partners of the normal branes and can be understood as sources of non-geometric backgrounds. Such backgrounds cannot be defined in terms of manifolds, instead these are described in terms of T(U)-folds [4,15,16], whose patches are glued by T(U)-duality transformations. At the level of field configurations this is realized as a monodromy when going around the point the exotic brane is placed [17].

It is convenient to label exotic branes in the same way as states of the 3D maximal supergravity are classified [3,17]. Hence, for any brane of string theory one adopts the notation *<sup>b</sup>*(*cr*...*c*1) *n* , where *b* + 1 gives the number of world-volume directions, *ci* denote the number of special (quadratic, cubic etc.) directions and *n* gives the power of the string coupling constant *gs* in tension of the brane. Such brane completely wrapped on a torus would have tension given by

$$M\left[b\_n^{(c\_r\dots c\_1)}\right] = \frac{R\_{i\_1}\dots R\_{i\_b}R\_{j\_1}^2\dots R\_{j\_{c\_1}}^2\dots}{\mathcal{g}\_s^n I\_s^{b+2c\_1+3c\_2+\dots(r+1)c\_r+1}},\tag{28}$$

where *Ri* denote radius of the *i*-th toroidal direction and *ls* is the string length. For example, for the NS5-brane, which has 6 world-volume directions, no special circles and whose tension scales as *g*<sup>−</sup><sup>2</sup> *s* , one would use the notation *NS*5 = 502 ≡ 52. Kaluza-Klein monopole is denoted as *KK*5 = 512 and has 6 worldvolume directions, one special circle corresponding to the Hopf fibre and its tension scales as *g* −2 *s* . In these notations duality symmetries of string theory act on such states as follows

$$T\_x: \quad R\_x \to \frac{l\_s^2}{R\_x}, \quad \mathbf{g}\_s \to \frac{l\_s}{R\_x} \mathbf{g}\_s; \qquad \mathbf{S}: \quad \mathbf{g}\_s \to \frac{1}{\mathbf{g}\_s}, \quad l\_s \to \mathbf{g}\_s^{\frac{1}{2}} l\_s. \tag{29}$$

The well-known example of a T-duality orbit containing exotic branes has been investigated in [18] (see also [19] for a review) and reads

$$5\_2^0 \quad \rightarrow \quad 5\_2^1 \quad \rightarrow \quad 5\_2^2 \quad \rightarrow \quad 5\_2^3 \quad \rightarrow \quad 5\_2^4. \tag{30}$$

The orbit starts with the NS5-brane, which is completely geometric and whose background can be consistently described both locally and globally in terms of the metric and the gauge field. Performing T-duality transformations along smeared transverse directions one obtains KK5-monopole 512 and then the exotic 522-brane. The background of the 522-brane cannot be described globally as it has non-trivial monodromy and is glued by T-duality. Going further along the orbit one recovers 532-branes, whose background is not well defined even locally, and 542-brane which is object of co-dimension-0. These branes, the corresponding backgrounds and their description in terms of T-duality covariant field theory will be considered in more details in Section 3.1.

Branes of M-theory completely wrapped on *d* compact direction with radii *Ri* are in correspondence with massive BPS states of maximal (11 − *d*)-dimensional supergravity, and can be labelled *b*(*cr*...*c*1) similarly to the branes of string theory. Tension of *<sup>b</sup>*(*cr*...*c*1)-brane, or equivalently mass of the corresponding state, is then given by

$$\mathcal{M}\left[b^{(c\_r\dots c\_1)}\right] = \frac{R\_{i\_1}\dots R\_{i\_b}R\_{j\_1}^2\dots R\_{j\_{c\_1}}^2\dots}{I\_{11}^{b+2c\_1+3c\_2+\dots(r+1)c\_r+1}},\tag{31}$$

where *l*11 is the 11-dimensional Planck mass. In these notations the M2-brane is denotes as 20 ≡ 2, the KK6-monopole is denoted as 61.

The exotic branes 53, 26 and 0(1,7) whose winding modes contribute counting for the *<sup>E</sup>*8(8) U-duality group, have three, six and 7 special Hopf-fiber-like direction. In addition, the 0(1,7) brane has one cubic special circle. Interpretation of cubic circles is more subtle that that of the quadratic ones, some discussion of these for branes of the Type II theories can be found in [20]. Windings modes for these branes correspond to components of the following mixed symmetry tensors

$$\begin{array}{llll} \mathbb{S}^3: & z\_{a\_1\ldots a\_8, b\_1 b\_2 b\_3} & \# = 56\\ \mathbb{S}^6: & z\_{a\_1\ldots a\_8, b\_1\ldots b\_6} & \# = 28\\ \mathbb{S}^{(1,\mathbb{T})}: & z\_{a\_1\ldots a\_8, b\_1\ldots b\_{\mathcal{T}}c\_1} & \# = 8 \end{array} \tag{32}$$

where indices to the right after commas must be equal to the those to the left, i.e., for the last line we have *c*1 = *b*1 = *a*1 and *b*'s = *a*'s. One immediately notices the reason these branes are the ones which are able to contribute the counting of winding for the *<sup>E</sup>*8(8) theory. Indeed, this U-duality group corresponds to *d* = 8 toroidal directions, hence 8 is the maximal number of antisymmetric indices one can have in a winding mode. Say for the 53 brane with 3 special directions this provides room for 5 longitudinal directions, while for the 0(1,7) one is left with none, hence a 0-brane.

It is important to note that although one includes windings of exotic branes into the counting for the *<sup>E</sup>*8(8) case, components of the generalized momentum still do not sum up to 248, the smallest possible irrep being rather 240. This cannot be cured by adding more exotic branes, since no other such brane is able to fit the 8-torus. More importantly, as it will be clear from analysis of the field spectrum of exceptional field theory, the problem is much more than just counting of extra coordinates, a huge part of which would be dropped anyway. Vector fields of the theory must be in the same irrep as the generalized momentum to add up to a covariant derivative, hence one needs extra 8 field to build up an irrep. This is indeed what happens in the *<sup>E</sup>*8(8) ExFT: one introduces extra tensor fields and imposes conditions on them similar to the section constraint [21]. For *<sup>E</sup>*9(9), which is an affine algebra with infinite-dimensional representations, one has to add infinitely many coordinates and fields. For larger U-duality groups the problem is even more complicated as first one encounters *E*10, which is an extension of the *E*8 by two imaginary roots, and then *E*11, which must encode timelike direction as well. Some progress in the construction of such infinitely dimensional extended spaces and the corresponding theories has been made in [22–25].

#### *2.3. Exceptional Field Theories*

Consider a general exceptional field theory defined on *D*-dimensional space-time parametrized by coordinates *<sup>x</sup>μ*, which we will call "external" and *N*-dimensional extended space, which we will call "internal" parametrized by X *M*. It is important to note, however, that no compactification is assumed in formulation of the theory. Toroidal reductions of supergravity described in the previous subsection have been used exclusively for counting winding modes. In the framework of exceptional field theories, toroidal backgrounds represent a solution, which preserves the maximal amount of U-duality. In this respect, toroidal backgrounds for ExFT are the same as Minkowski for General Relativity. In what follows, all fields are allowed to depend on the whole set of coordinates (*xμ*, X *<sup>M</sup>*).

On this space the following field content is defined: external metric *gμν*, *N* vector fields *<sup>A</sup>μM*, generalized metric M*MN* and various external tensors <sup>B</sup>*μ*1...*μr α* transforming under an irrep of *G* the U-duality group. Generalized metric parametrized the coset space *G*/*K*, where *K* is the maximal compact subgroup of *G*. For supersymmetric theories one in addition defines fermions, transforming under an irrep of *K*.

Locally geometry of the extended space parametrized by X *M* is represented by the generalized Lie derivative L Λ, which defines generalized diffeomorphism transformations of generalized tensors along a generalized vector Λ *M*. Since, the vector Λ *M* = Λ *<sup>M</sup>*(*<sup>x</sup>*, X) depends both on the external and internal coordinates, partial derivative *∂μ* of a generalized tensor does not transform covariantly. To fix that one introduces long derivative following the usual approach of the Yang-Mills theory

$$D\_{\mu} = \partial\_{\mu} - \mathcal{L}\_{A\_{\mu}}.\tag{33}$$

Hence, the vector fields *<sup>A</sup>μ<sup>M</sup>* play the role of gauge connection in the theory. The corresponding field strength is defined in the usual way

$$
\mathbb{E}\left[D\_{\mu\nu}D\_{\nu}\right] = -\mathcal{L}\_{F\_{\mu\nu}}.\tag{34}
$$

One however faces a subtlety here noticing that such defined field strength *<sup>F</sup>μν<sup>M</sup>* is not a generalized tensor. This situation is familiar from the construction of maximal gauged supergravities, where one has to add *p*-form gauge field to a *p*-form field strength to build covariant expressions. Eventually, this leads to tensor hierarchy of the theory [26]. Hence, one defines

$$\mathcal{F}^{M}\_{\mu\nu} = F\_{\mu\nu}{}^{M} + \mathcal{Y}^{MN}\_{KL} \partial\_N B\_{\mu\nu}{}^{KL} \,, \tag{35}$$

using that LF*μν* ≡ <sup>L</sup>*<sup>F</sup>μν* up to the section constraint. In general, one shows that generalized vectors of the form <sup>Λ</sup>*M*0 = *YMN KL <sup>∂</sup>NχKL* for any *χKL* do not induce generalized diffeomorphisms when the section constraint is satisfied (for details see [27]). Hence, the 2-forms of the theory came from reduction of the 3-form of 11-dimensional supergravity are utulized in the covariant field strength.

If the 2-forms are dynamical, one needs to construct a covariant field strength for them, which will enter the fully covariant Lagrangian. For that, one considers Bianchi identity for the 2-form field strength, which can be written as

$$3D\_{\left[\mu\right.} \mathcal{F}\_{\nu\rho\right]}{}^{M} = \mathcal{Y}\_{KL}^{MN} \partial\_{N} H\_{\mu\nu\rho}{}^{KL}.\tag{36}$$

As before, such recovered field strength *<sup>H</sup>μνρKL* is covariant only when contracted with the Y-tensor, and hence needs additional contributions. One invokes 3-form potential *<sup>C</sup>μνρM*,*KL*, for whose field strength one needs a 4-form and so on. This tower of *p*-forms and their field strengths is called tensor hierarchy of exceptional field theories and drops to that of maximal gauged supergravities upon the Scherk-Schwarz reduction ansatz (see below). In principle, the hierarchy can go up to the top form; however it ends much earlier since at some point one no longer needs to construct ye<sup>t</sup> another field strength since the corresponding gauge field is non-dynamical and does not enter the Lagrangian. Although the *E*7 ExFT is more suitable for phenomenological applications, it has short tensor hierarchy and issues with self-duality. Hence, let us consider the construction of the SL(5) ExFT in more detail to illustrate the general idea.

Covariant field strength for the gauge field A*μmn* that is recovered from the commutator of long derivatives and then shifted by a derivative of <sup>B</sup>*μν<sup>m</sup>*, takes the following form [28]

$$\mathcal{F}^{mn}\_{\mu\nu} = 2\partial\_{[\mu}\mathcal{A}^{mn}\_{\nu]} - [\mathcal{A}\_{\mu}, \mathcal{A}\_{\nu}]^{mn}\_{\gets} + \frac{1}{4}\epsilon^{mnkl\,p}\partial\_{kl}\mathcal{B}\_{\mu\nu\overline{p}\,\nu} \tag{37}$$

where  *mnklp* is the fully antisymmetric invariant tensor of SL(5). Explicit check shows that under generalized transformations of the fundamental fields A*μmn* and <sup>B</sup>*μνm* such defined field strength transforms covariantly, i.e., *<sup>δ</sup>*ΛF*μνmn* = <sup>L</sup>ΛF*μνmn*.

Bianchi identity for the 2-form field strength has non-trivial RHS and which can be written as a full derivative of a tensor <sup>F</sup>*μνρm*

$$\mathcal{G}\mathcal{D}\_{\left[\mu\right.}\mathcal{F}\_{\nu\rho\right]}{}^{mn} = -\frac{1}{16}\epsilon^{imnkl}\partial\_{kl}\mathcal{F}\_{\mu\nu\rho i}.\tag{38}$$

This tensor however has a freedom in adding of terms of the type

$$
\partial\_{mm}\mathcal{C}\_{\mu\nu\rho}{}^{m}.\tag{39}
$$

Indeed, in the Bianchi identity such term will give  *imnkl∂kl∂ij*C*μνρ j* ≡  *imnkl∂*[*kl∂ij*]C*μνρ j*, which vanishes upon the section constraint of the theory

$$
\epsilon^{imkl}\partial\_{mn}\bullet\partial\_{kl}\bullet = 0.\tag{40}
$$

To match tensor hierarchy of the maximal *D* = 7 gauged supergravity [29] one fixes a coefficient in front of the term and the covariant 3-form field strength reads

$$\mathcal{F}\_{\mu\nu\rho m} = 3\mathcal{D}\_{[\mu}\mathcal{B}\_{\nu\rho]m} + \frac{3}{2}\epsilon\_{m\nu\rho\tau s} \left( A\_{[\mu}^{pq}\partial\_{\nu}A\_{\rho]}^{rs} - \frac{1}{3}[A\_{[\mu},A\_{\nu}]\_{E}{}^{pq}A\_{\rho]}{}^{rs} \right) - \frac{1}{4}\partial\_{mn}\mathcal{C}\_{\mu\nu\rho}{}^{n} . \tag{41}$$

Now one notices by looking at the Figure 2 that there are too little degrees of freedom to compose both the 2-form and the 3-form gauge potentials. On the other hand, according to the prescription of [11] the three form *<sup>C</sup>μνρ* should be dualized into a two form to complete the irrep **5** of SL(5). Speaking about covariant theories, one would like to keep information about the 3-form, which is done by taking into account an alternative prescription, which is to dualize the 2-forms *<sup>B</sup>μνa* into 3-forms to complete the irrep **5¯** of SL(5). Hence, one ends up with the same degrees of freedom encoded in the fields <sup>B</sup>*μνm* and <sup>C</sup>*μνρ<sup>m</sup>*. The duality relation between these potentials can be written in the following covariant form

$$m^{mn}\mathcal{F}^{\mu\nu\rho}{}\_n - \frac{1}{4!} \epsilon^{\mu\nu\rho\lambda\sigma\tau\kappa} \mathcal{F}\_{\lambda\sigma\tau\kappa}{}^m = 0,\tag{42}$$

where  *μνρσκλτ* is the Levi-Civita tensor in the external 7 directions and *mmn* is the inverse of the generalized metric *mmn*.

**Figure 2.** Reduction of 11D fields into seven dimensions, dualization into the lowest possible rank tensor and recollection into SL(5) multiplets. Dashed lines correspond to an alternative combination of the 2- and 3-forms into the 3-form *<sup>C</sup>μνρ<sup>m</sup>* dual to *<sup>B</sup>μνm* in 7 dimensions (see the text). Here the small Latin indices *m*, *n*, *k*, ... label the fundamental irrep **5** of SL(5) and small Latin indices from the beginning of the alphabet label the fundamental irrep **4** of GL(4).

Since the 3-form field strength contains the same information as the 4-form field strength, the latter is considered to be non-physical and is not included into the Lagrangian. It appears only in the corresponding Bianchi identity

$$4\mathcal{D}\_{\left[\mu\right.} \mathcal{F}\_{\nu\rho\sigma\right]m} = 6\varepsilon\_{mp\eta\sigma\kappa} \mathcal{F}\_{\left[\mu\nu\right.} \mathcal{F}\_{\left[\mu\nu\right.} \mathcal{F}\_{\rho\sigma\rceil} \left]^{rs} + \partial\_{nm} \mathcal{F}\_{\mu\nu\rho\sigma} \right. \tag{43}$$

and is needed to ensure invariance of the Lagrangian with respect to external diffeomorphisms. Note that the 3-form gauge potential containing in the 4-form field strength has been already used for construction of the 3-form covariant field strength (see Figure 2).

Lagrangian of the theory is schematically the same as the one for the maximal *D* = 7 gauged supergravity

$$\begin{split} \mathcal{L}\_{\text{EFT}} &= \mathcal{L}\_{\text{EH}}(\vec{\mathcal{R}}) + \mathcal{L}\_{\text{sc}}(\mathcal{D}\_{\mu}m\_{\text{kl}}) + \mathcal{L}\_{V}(\mathcal{F}\_{\mu\nu}^{\text{mm}}) + \mathcal{L}\_{\text{T}}(\mathcal{F}\_{\mu\nu\rho\text{ m}}) \\ &+ \mathcal{L}\_{\text{top}} - V(m\_{\text{kl}}, \mathbf{g}\_{\text{fl}}). \end{split} \tag{44}$$

Here the kinetic part is given by the Einstein-Hilbert term, scalar kinetic term, and vector and tensor kinetic terms respectively. In addition, one includes the "scalar potential" *V*, which contains only terms with derivatives *∂mn* along extended coordinates, and the topological term <sup>L</sup>*top* which does not contain contractions with the metric. Explicitly the expressions read

$$\begin{split} \mathcal{L}\_{EH} &= \epsilon \mathcal{C}\_{\mu}^{H} \mathcal{R}\_{\mu\nu} \,^{a\beta} \,, \\ \mathcal{L}\_{\mathfrak{s}\mathfrak{c}} &= \frac{1}{12} \varepsilon \mathcal{g}^{\mu\nu} \mathcal{D}\_{\mu} \mathcal{M}\_{MN} \mathcal{D}\_{\nu} \mathcal{M}^{MN} \,, \\ \mathcal{L}\_{V} &= -\frac{1}{4} \varepsilon \mathcal{M}\_{MN} \mathcal{F}\_{\mu\nu} \,^{M} \mathcal{F}\_{\mu\nu} \,^{N} = -\frac{1}{8} \varepsilon m\_{mk} m\_{n} \mathcal{F}\_{\mu\nu} \,^{mn} \mathcal{F}^{\mu\nu} \,^{M} \,, \\ \mathcal{L}\_{T} &= -\frac{1}{3 \cdot (16)^{2}} \varepsilon m^{mn} \mathcal{F}\_{\mu\nu\rho} \mathcal{F}^{\mu\nu\rho} \,^{a} \\ V &= -\frac{1}{13} \mathcal{M}^{MN} \partial\_{M} \mathcal{M}^{KL} \partial\_{N} \mathcal{M}\_{KL} + \frac{1}{2} \mathcal{M}^{MN} \partial\_{M} \mathcal{M}^{KL} \partial\_{L} \mathcal{M}\_{NK} \\ &- \frac{1}{2} (\mathcal{g}^{-1} \partial\_{M} \mathcal{g}) \partial\_{N} \mathcal{M}^{MN} - \frac{1}{4} \mathcal{M}^{MN} (\mathcal{g}^{-1} \partial\_{M} \mathcal{g}) (\mathcal{g}^{-1} \partial\_{N} \mathcal{g}) - \frac{1}{4} \mathcal{M}^{MN} \partial\_{M} \mathcal{g}^{\mu\nu} \partial\_{N} \mathcal{g}\_{\mu\nu} \,^{M} \end{split} \tag{45}$$

Here the modified Riemann curvature *R* ˆ *μνab* = *<sup>R</sup>μναβ* + <sup>F</sup>*μνMeρα∂M<sup>e</sup>ρβ* transforms covariantly under both generalized and external diffeomorphisms. The potential *V* is written in terms of the 10×10 generalized metric *MMN* which is related to the 5 × 5 metric *mmn* by

$$M\_{mn\,kl} = m\_{mk}m\_{nl} - m\_{ml}m\_{nk\,\prime} \tag{46}$$

i.e., each capital index *M* labelling the **10** is represented by an antisymmetric pair of indices [*mn*] each labelling the fundamental **5** of SL(5). In addition, one halves each contraction of capital indices when turning to pairs to avoid extra counting.

The topological part cannot be written in a covariant form, precisely as in the gauged supergravity case; however its variation can

$$\delta \mathcal{L}\_{\text{top}} = \frac{1}{16 \cdot 4!} \varepsilon^{\mu\nu\rho\lambda\sigma\tau\kappa} \left[ \mathcal{F}\_{\mu\nu\rho\lambda}{}^{\text{m}} \partial\_{\text{mn}} \Lambda \mathcal{C}\_{\sigma\tau\kappa}{}^{\text{n}} + 6 \mathcal{F}\_{\mu\nu}{}^{\text{mu}} \mathcal{F}\_{\rho\lambda\sigma\text{m}} \Delta B\_{\tau\kappa\text{n}} - 2 \mathcal{F}\_{\mu\nu\rho\mu} \mathcal{F}\_{\lambda\sigma\tau\kappa} \delta A\_{\kappa}{}^{\text{mu}} \right]. \tag{47}$$

Transformations Δ of the fields denote external diffeomorphisms parametrized by a vector *ξμ*, generalized diffeomorphisms parametrized by a generalized vector Λ*mn* and gauge transformations of the fields A*μmn*, <sup>B</sup>*μν<sup>m</sup>*, <sup>C</sup>*μνρ<sup>m</sup>*:

$$\begin{split} \delta \varepsilon^{a}\_{\mu} &= \xi^{\mu} \mathcal{D}\_{\nu} \varepsilon^{a}\_{\mu} + \mathcal{D}\_{\mu} \xi^{\nu} \varepsilon^{a}\_{\nu} \\ \delta \mathcal{M}\_{\text{MN}} &= \xi^{\mu} \mathcal{D}\_{\mu} \mathcal{M}\_{\text{MN}} \\ \delta A\_{\mu}{}^{M} &= \xi^{\nu} \mathcal{F}\_{\nu\mu}{}^{M} + \mathcal{M}^{\text{MN}} g\_{\mu\nu} \partial\_{N} \xi^{\nu} + \mathcal{D}\_{\mu} \Lambda^{\text{mn}} + \frac{1}{16} \varepsilon^{imkl} \partial\_{kl} \Xi\_{\mu i} \\ \Delta B\_{\mu i\dot{\imath}} &= \xi^{\rho} \mathcal{F}\_{\rho\mu\dot{\imath}i} + 2 \mathcal{D}\_{[\mu} \Xi\_{\nu]i} - 2 \varepsilon\_{imnpq} \Lambda^{mn} \mathcal{F}\_{\mu\nu}{}^{pq} - \partial\_{m} \Psi\_{\mu\nu}{}^{m} \\ \Delta \mathcal{L}\_{\mu\nu p}{}^{m} &= -\frac{1}{3!} \varepsilon \varepsilon\_{\mu\nu\rho\nu\lambda\dot{\imath}} \xi^{\rho} m^{mn} \mathcal{F}^{\kappa\lambda\tau}{}\_{n} + 3 \mathcal{D}\_{[\mu} \Psi\_{\nu]}{}^{m} + 3 \mathcal{F}\_{[\mu\nu}{}^{mn} \Xi\_{\rho]n} + \Lambda^{mn} \mathcal{F}\_{\mu\nu pn}. \end{split} \tag{48}$$

The amazing feature of exceptional field theories observed already in [30] is that the form of the Lagrangian is completely fixed by demanding invariance with respect to the above transformations. In contrast, in gauged supergravities one has to impose supersymmetry to fix all coefficients, although the general structure of the Lagrangian is completely the same. The non-trivial check is that supersymmetry works well for the Lagrangian of exceptional field theory fixed in such a way. This has been checked explicitly for *E*7 and *E*6 in [31,32], and is expected to work for all other cases as well.

#### *2.4. The Section Constraint*

Exceptional field theories are formulated on a space-time parametrized by *d* external (conventional) coordinates *xμ* and *D* "internal" extended coordinates X*M*. Since no reduction is assumed, these are theories on a *d* + *D*-dimensional space-time, which always has dimension greater than 11. To reduce the amount of degrees of freedom, and to ensure consistency of algebra of generalized diffeomorphism one has to impose condition, the section constraint, which is

$$
\gamma\_{KL}^{MN} \partial\_M \bullet \partial\_N \bullet \tag{49}
$$

where • stands for any field or combination of fields. Y-tensors for different U-duality groups are listed in Table 1.

This condition restricts dependence of fields of the theory on the extended coordinates and can be solved in various ways. Consider as an example the *E*7 theory for which the condition is

$$\begin{aligned} \, \, \Omega^{MN} \partial\_M \bullet \, \partial\_N &= 0, \\ \, t\_a \, ^{MN} \partial\_M \bullet \, \partial\_N &= 0 \end{aligned} \tag{50}$$

where Ω*MN* is the invariant symplectic form of E7 < Sp(56) and *tα* are generators of the group. In [33] it has been shown that this condition has precisely two algebraic types of solutions: corresponding to embeddings of the D = 11 maximal supergravity and of Type IIB supergravity. All other follow from this two. In addition, there is a special way to relax this constraint by imposing the so-called Scherk-Schwarz ansatz.

Consider first the algebraic solution of the section constraint corresponding to embedding of the 11D maximal supergravity. In this case, one decomposes the fundamental **56** of E7 under the action of the *GL*(7) subgroup as

$$\mathbf{56} \rightarrow \mathbf{7}\_{+\mathbf{3}} + \mathbf{21}'\_{+\mathbf{1}} + \mathbf{21}\_{-\mathbf{1}} + \mathbf{7}'\_{-\mathbf{3}'} \tag{51}$$

where subscripts denote the GL(1) weight. For the coordinates X*<sup>M</sup>* this implies

$$\mathbb{X}^{M} = (\mathbf{x}^{m}, \mathbf{y}\_{mn}, \mathbf{y}^{nm}, \mathbf{x}\_{m}). \tag{52}$$

Here and further in this subsection

$$\begin{aligned} \text{M}, \text{N}, \text{K}, \dots &= 1, \dots, 56, \quad \text{label the } \mathbf{56} \text{ of } \text{E}\_7, \\ \text{m}, n, k \dots &= 1, \dots, 7, \quad \text{label the } \mathbf{7} \text{ of } \text{SL}(7), \\ a, b, c, d, \dots &= 1, \dots, 6, \quad \text{label the } \mathbf{6} \text{ of } \text{SL}(6), \\ a, \beta, \dots &= 1, 2, \quad \text{label the } \mathbf{2} \text{ of } \text{SL}(2). \end{aligned} \tag{53}$$

The section constraint then can be satisfied by imposing

$$\frac{\partial}{\partial y\_{mn}} = 0, \quad \frac{\partial}{\partial \tilde{y}^{mn}} = 0, \quad \frac{\partial}{\partial \tilde{x}\_{\text{ill}}} = 0,\tag{54}$$

hence fields depend only on 7 coordinates *x<sup>m</sup>* and 4 coordinates *<sup>x</sup>μ*, in total 11. Field content of the E7 theory decomposed under the GL(7) subgroup apparently fits the field content of the 11 dimensional supergravity in the7+4 split form, as the former was constructed from the latter initially. Type IIA supergravity is obtain from this theory by further S1 reduction in the usual way.

The alternative solution of the section constraint is associated with decomposition under the embedding E7 ←- GL(6) × SL(2), which implies for the fundamental irrep

$$\mathbf{56} \to (6,1)\_{+2} + (6',2)\_{+1} + (20,1)\_0 + (6,2)\_{-1} + (6',1)\_{-2}.\tag{55}$$

For components of the extended coordinates this reads

$$\mathbb{X}^{M} = (\mathfrak{x}^{a}, y\_{a\alpha}, y\_{abc}, \mathfrak{y}^{a\alpha}, \mathfrak{x}\_{a}), \tag{56}$$

and the field content will be decomposed accordingly, The SL(2) subgroup transforming subsets of dual coordinates correspond to S-duality symmetry of Type IIB supergravity. The section constraint implies

$$
\frac{\partial}{\partial y\_{an}} = 0, \quad \frac{\partial}{\partial y\_{abc}} = 0, \quad \frac{\partial}{\partial \bar{y}^{an}} = 0, \quad \frac{\partial}{\partial \bar{x}\_a} = 0,\tag{57}
$$

and the fields depend on 6 coordinates *x<sup>a</sup>* and 4 coordinates *<sup>x</sup>μ*, in total 10. After reduction, the SL(2) S-duality symmetry can be kept manifestly at the cost of the full 10-dimensional Lorenz symmetry. To restore the latter, one has to break the former.

A special solution to the section constraint trivially embedded into the above two classes is *∂M* = 0, i.e., all fields do not depend on extended coordinates. This corresponds simply to *d*-dimensional maximal ungauged supergravity, i.e., a reduction of the 11-dimensional supergravity on a torus T11−*d*. These are known to allow deformations, gaugings, producing a class of theories corresponding to various reductions of the initial 11-dimensional theory [26]. Exceptional field theories are able to reproduce such gauged supergravities as well under a special ansatz, which relaxes the differential section condition to an algebraic constraint. These will be considered in more detail in Section 3.1.

#### *2.5. Double Field Theory*

Double field theory, which is a T-duality covariant formulation of supergravity, also fits the above scheme. The duality group is O(n,n) with *n* + *d* = 10, the Y-tensor then becomes *YMN KL* = *<sup>η</sup>MNηKL*, where *M*, *N*, *K*, *L* = 1, ... , 2*n* for any *n*. Since algebraic structure of the theory does not depend on the number of "internal" dimensions, it is natural to extend all 10 coordinates of Type II supergravity. The generalized metric then

$$M\_{MN} \in \frac{\mathcal{O}(10, 10)}{\mathcal{O}(1, 9) \times \mathcal{O}(9, 1)} \,\prime \tag{58}$$

and the rest of the construction is repeated identically.

Important remark here concerns the section constraint, which is

$$
\eta^{MN}\partial\_M\bullet\partial\_N\bullet = \partial\_m\bullet\tilde{\partial}^m\bullet = 0,\tag{59}
$$

where we introduce notation X *M* = (*xm*, *<sup>x</sup>*˜*m*) for extended coordinates. The above implies that for any pair of a normal coordinate *x*<sup>∗</sup> and the corresponding dual coordinate *<sup>x</sup>*˜∗ fields are allowed do depend on one or another, but not on both. The choices are related by a T-duality transformation precisely as in the Duff's procedure

$$T\_\* \colon \mathfrak{x}^\* \longleftrightarrow \mathfrak{x}\_\* \,. \tag{60}$$

Hence, such defined T-duality means that one translates all upper indices ∗ into lower indices ∗, and the coordinates *<sup>x</sup>*˜∗ becomes normal geometric coordinate in the new T-duality frame. Note however a different possibility, where one simply replaces dependence of background fields on *x*<sup>∗</sup> by *<sup>x</sup>*˜∗, still counting the latter as a non-geometric dual coordinate. This will turn a solution of supergravity equations of motion into a proper string background, which however (i) does not solve e.o.m.s, (ii) is non-geometric. This way DFT and ExFT allow addressing exotic branes and the corresponding non-geometric backgrounds. See further Section 3.2 for more details.
