**3. Applications**

## *3.1. Non-Geometric Compactifications*

Choosing a solution of the section constraint of exceptional field theory can be understood as a dimensional reduction as it drops dependence of fields of the theory on a subset of coordinates. E.g., the trivial solution *∂M*• = 0 when all fields depend only on *D* external coordinates corresponds to reduction of the 11-dimensional supergravity on a 11 − *D*-torus. Other toroidal reductions can as well be obtained in such a way by choosing a smaller subset of the extended coordinates to be dropped. However, exceptional field theory is able to incorporate more general reductions, in fact all parametrized in terms of embedding tensor.

Taking the general idea of twisted dimensional reduction of Scherk and Schwarz [34] one introduces the following generalized Scherk-Schwarz ansatz

$$V^M(\mathbf{x}, \mathbb{X}) = \mathcal{U}^M{}\_A(\mathbb{X}) V^A(\mathbf{x}),\tag{61}$$

where *UMA*(X) is usually referred to as the twist matrix. The ansatz tells that all information about dependence on the extended coordinates X*<sup>M</sup>* is contained in the twist matrix. Inserting this into the definition of generalised Lie derivative one obtains

$$
\mathcal{L}\_{\Lambda}V^{M} = \mathcal{U}^{M}{}\_{A}(\mathbb{X}) X\_{B\mathbb{C}}{}^{A}\Lambda^{A}(\mathbf{x}) V^{B}(\mathbf{x}),\tag{62}
$$

which can be thought of as a definition of the tensor *XAB<sup>C</sup>*

$$X\_{AB}{}^{\mathbb{C}} \equiv 2I I\_M{}^{\mathbb{C}} \mathcal{U}\_{[A}{}^{N} \partial\_{N} \mathcal{U}\_{B]}{}^{M} + Y\_{EB}^{CD} \mathcal{U}\_{M}{}^{E} \mathcal{U}\_{D}{}^{N} \partial\_{N} \mathcal{U}\_{A}{}^{M} \,. \tag{63}$$

where *UMCUCN* = *<sup>δ</sup>MN*. In principle, this tensor depends on X*<sup>M</sup>* as it is constructed out of twist matrices and their derivatives; however, for now we will assume *XAB<sup>C</sup>* = const. Certainly this implies restrictions on the twist matrices which will be discussed in a moment.

Upon the generalised Scherk-Schwarz ansatz the closure constraint (23) and the condition for *XAB<sup>C</sup>* to transform covariantly boil down to the following simple algebraic constraint

$$[X\_{A\prime}X\_B] = -X\_{AB}{}^{\mathbb{C}}X\_{\mathbb{C}}.\tag{64}$$

Here we understand *XAB<sup>C</sup>* as a matrix labelled by *A*. The above looks as a commutation relation for an algebra with generators *XA*; however, since *XAB<sup>C</sup>* is not necessarily antisymmetric, such simple interpretation does not work. Instead, one recognizes here the structure of gauged supergravities, with *XAB<sup>C</sup>* being the embedding tensor. Hence, the main line of the subsection is that Scherk-Schwarz reduction of exceptional field theories replaces the differential section constraint by the algebraic condition (64), which has precisely the same form as the quadratic constraint of gauged supergravity. Moreover, performing the reduction at the level of the action one reproduces precisely action of the corresponding *D*-dimensional gauged supergravity. This has been shown first for DFT in [35] and then for exceptional field theories SL(5), SO(5,5), *E*6, *E*7 and the enhanced O(d,d) exceptional field theory in [36–39]. Let us now briefly describe structure of gauged supergravities and their relation to generalized Scherk-Shwarz reductions.

Gauged supergravities most straightforwardly can be described in the so-called embedding tensor formalism first developed in the context of three dimensional theories [40,41] and then constructed for other maximal supergravities (for review see [26,42,43]). When dimensionally reduced on a *d*-torus 11-dimensional supergravity produces maximally supersymmetric theory in *D* = 11 − *d* dimensions, which contains *nV* vector fields <sup>A</sup>*μA*, with *A* = 1, ... , *nV*. The vector fields descent from the metric

and the 3-form C-field in 11 dimensions precisely in the same way as the generalized vector fields of exceptional field theory (see Figures 1 and 2). For toroidal reductions the resulting theory is abelian with the gauge group *U*(1)*nV* and gauge transformations

$$
\delta\_\Lambda \mathcal{A}\_\mu{}^A = \partial\_\mu \Lambda^A. \tag{65}
$$

In addition, the lower dimensional theory has U-duality symmetry group *G* and the vector multiplet transforms under an irrep R*V* of dimension dim R*V* = *nV*. Reductions on more complicated manifolds endowed with torsion and/or curvature, and reductions in the presence of fluxes of gauge fields result in theories with less symmetry, and with vector multiplets belonging to adjoint representation of a non-abelian gauge group. Such reductions correspond to the diagonal arrow on Figure 3, while toroidal reductions correspond to the vertical arrow. Hence, it is natural to deform the abelian maximal theory introducing non-abelian interactions between the *nV* vector fields, preserving supersymmetry and local symmetries of the theory. A self-consistent algorithm for such a procedure is based on the notion of embedding tensor Θ, which defines embedding of the desired local gauge group *G* into the full global U-duality group *G*

$$
\Theta \colon G \to G'. \tag{66}
$$

One writes deformation of the gauge transformation rule for vector multiplets as follows

$$
\delta\_{\Lambda} \mathcal{A}\_{\mu}{}^{A} = \partial\_{\mu} \Lambda^{A} - \lg X\_{BC}{}^{A} \Lambda^{B} \mathcal{A}\_{\mu}{}^{C}, \tag{67}
$$

where *g* is the deformation parameter. The "structure constants" *XAB<sup>C</sup>* can be written in terms of the embedding tensor as

$$X\_{AB}{}^{\mathbb{C}} = Z\_{AB}{}^{\mathbb{C}} + \mathbb{X}\_{AB}{}^{X} = \Theta\_A{}^{a} t\_{aB}{}^{\mathbb{C}},\tag{68}$$

where {*<sup>t</sup>α*} = bas g is basis of generators of the global U-duality group *G*, and *Z* and *X*ˆ are symmetric and antisymmetric in {*AB*} respectively. Hence, the embedding tensor indeed selects a subset of these generators to construct the constants *XAB<sup>C</sup>*, which define the non-abelian gauge transformation.

**Figure 3.** Relations between toroidal reductions of N = 1 *D* = 11 supergravity, gaugings and more complicated dimensional reductions involving geometric and non-geometric fluxes.

Certainly, such a deformation of the ungauged *D*-dimensional theory cannot be done in an arbitrary way and has rather to satisfy certain constraints. The first one is the linear constraint and comes from the condition that the resulting deformed equations of motion are still invariant under the full set of supersymmetries, i.e., the deformed gauged supergravity is a maximally supersymmetric

theory. The linear constraint projects certain representations of *G* in the decomposition of the embedding tensor Θ*A<sup>α</sup>* ∈ R*nV* ⊗ adj g, which can be schematically written as

$$
\mathbb{P}\,\Theta = 0.\tag{69}
$$

As an example, one may consider the *D* = 4 theory constructed in [13,44], for which *G* = R<sup>+</sup> × *E*7, R*nv* = **56**, adj g = 133 and

$$
\Theta\_A{}^a \in \mathbf{56} \otimes \mathbf{133} = \mathbf{56} + \mathbf{912} + \mathbf{6480}.\tag{70}
$$

The linear constraint tells that for the theory to be maximally supersymmetric the embedding tensor cannot have the **6480** part

$$\mathbb{P}\_{\mathsf{6480}} \oplus = 0.\tag{71}$$

Please note that the duality group to be gauged contains a R<sup>+</sup> part, which correspond to global rescalings of the external metric. After the gauging procedure the corresponding generators correspond to the so-called trombone gauging. Certainly, theories with non-zero trombone part of the embedding tensor cannot be written in terms of an action, as rescalings are symmetries of equations of motion, rather than the action [44].

In addition to the linear constraint the embedding tensor must respect the so-called quadratic constraint, which follows from the condition of covariancy of the tensor under global transformations. These are nothing but (64)

$$
\mathbb{E}\left[X\_{A\prime}X\_{B}\right] = -X\_{AB}\,^{\mathbb{C}}X\_{\mathbb{C}}.\tag{72}
$$

Note that the LHS does not have the part symmetric in {*<sup>A</sup>*, *<sup>B</sup>*}, while the tensor *XAB<sup>C</sup>* has no symmetry in the lower indices in general. Hence, the above implies constraint on the symmetric part *ZABCXCDE* = 0. This condition is essential for the Jacobi identity for gauge transformations. Indeed, one discovers that the Jacobiator of gauge transformations has the conventional form

$$[\delta\_{\Lambda\_1}, \delta\_{\Lambda\_2}, \delta\_{\Lambda\_3}]V^A = -3\hat{\mathcal{X}}\_{[BC}{}^G \hat{\mathcal{X}}\_{D]G}{}^E X\_{EF}{}^A \Lambda\_1 \Lambda\_2 \Lambda\_3 V^F. \tag{73}$$

However, the commutation relations (64) restricted for the antisymmetric part only do imply the RHS of the above expression to vanish, which is usually the case for a conventional Lie algebraic structure constants. Instead one again finds the symmetric part

$$\left[3\hat{\mathcal{X}}\_{[\rm BC}\,^G\hat{\mathcal{X}}\_{D]G}\right]^E = Z\_{G[B}{}^E X\_{CD]}{}^G. \tag{74}$$

hence, the Jacobi identity is satisfied upon *ZABCXC* = 0 similarly to the closure constraint.

Such rich algebraic structure of the deformations means that one is not simply dealing with a single gauge group, but rather with a set of gauge groups inside the full duality group *G*, defined by choosing the embedding tensor. Gauged supergravities for different dimensions have been constructed in the series of papers [13,29,44–47] which contain detailed analysis of the constraint briefly described above, construction of the corresponding effective action including the fermionic sector and analysis of some examples.

Solving the linear and quadratic constraints for the embedding tensor one ends up with a theory which can be in principle obtained as a dimensional reduction of the 11-dimensional supergravity on some manifold. For example, a particular choice of gaugings for the *D* = 4 theory correspond to embedding Θ : *G* → SO(8), which gives reduction on a S<sup>7</sup> sphere considered in [48,49]. In this case, components of the embedding tensor are equal to the integral value of the flux of the 7-form field strength on the sphere.

Such gaugings, which can be interpreted in terms of dimensional reductions are called geometric and are said to have higher dimensional origin. However, the full set of solutions to the quadratic and linear constraint contains gaugings which do not have apparent higher dimensional origin, moreover one can show that some cannot have any [50]. Although these still define consistent theories

in lower dimensions and give masses and couplings for lower dimensional fields, one cannot come up with a dimensional reduction scheme starting from the 11D supergravity and ending up with these theories.

At this point, one turns back to the beginning of this subsection, where Scherk-Schwarz reduction of exceptional field theories have been shown to produce algebraic structures of the same type as those of gauged supergravities. In principle, since any gauging is given as an expression written in terms of twist matrices and their derivatives, as in (63), one may hope that it is possible to solve these equations and present twist matrices explicitly for each type of gauging. This has been shown to be true for *D* = 7 half-maximal gaugings and *D* = 8 maximal gaugings in [50], where full classification of the corresponding twist matrices was obtained. The most outstanding output of such analysis is that the so-called non-geometric gaugings, i.e., those which do not have higher standard dimensional origin, can be constructed out of generalized twist matrices. Moreover, the set of non-geometric gaugings itself is divided into normal and genuine non-geometric subsets. The former is defined to belong to a U-duality orbit of a geometric gauging, and hence the theory can be rotated to a frame, where no non-geometry is present. On the other hand, the latter do not belong to such an orbit and hence the theory is always non-geometric. At least for some examples, genuine non-geometric gaugings were shown to descent from twist matrices which *break the section constraint*.

At this point it is suggestive to return to the constraint (64) and recall that for that to satisfy one does not need to impose any conditions on dependence of twist matrices on the extended coordinates. Hence, in principle it is possible to break the section constraint, while satisfying the quadratic constraint that has been shown explicitly in [50]. Certainly, such twist matrices cannot be understood as defining a reduction of the conventional supergravity. Whether such configurations can be understood as proper string backgrounds is not clear, as all duality-covariant setups so far imply the section condition. However, genuine non-geometric gaugings look very promising for cosmological model building as these provide extra parameters in scalar potential of the lower dimensional theory, which may help to stabilize its moduli. Indeed, if one considers a setup with geometric gaugings which fails to stabilize certain subset of scalar moduli, any gaugings belonging to duality orbits of the initial ones would also fail to do so, since the physics must be duality invariant. In contrast, genuine non-geometric gaugings do not suffer from such constraints and hence can enhance the realm of possible models. This approach has been taken to analyse cosmological implications of exceptional field theories in application to some toy-model examples in [51–55].

#### *3.2. Exotic Brane Backgrounds*

Geometric gaugings of lower dimensional supergravities are equal to integral values of fluxes on the internal manifold, and hence, naturally acquire interpretation in terms of branes. Wrapped around cycles of the internal space, these source fluxes of gauge *p*-form fields, which then are subject of the usual Dirac quantisation condition (see e.g., [56] for review). Similarly, non-geometric fluxes can be interpreted as integrated values of field strengths of mixed symmetry potentials sourced by exotic branes. Algebraic-wise one finds good correspondence between mixed allowed symmetry potentials of lower dimensional supergravity (equivalently, exotic branes) and gaugings [57–59]. However, the conventional supergravity is known to be unable to properly describe backgrounds of such exotic branes and to provide a technique for calculating the corresponding non-geometric fluxes.

The well-known example comes from the T-duality orbit of the NS5-brane, starting with the *H*-flux [18,19]. This is known to contain non-geometric Q- and R-fluxes via the relation

$$H\_{\rm abc} \quad \xleftarrow{T\_a} \quad f\_{ab}\xarrow \quad \xleftarrow{T\_b} \quad Q\_a^{\rm bc} \quad \xleftarrow{T\_c} \quad R^{\rm abc},\tag{75}$$

where *T*∗ denotes T-duality transformation in the corresponding isometry direction. Since in the conventional supergravity T-duality transformation is defined only along a Killing direction and T-duality along the world-volume direction of the NS5-brane do not change the background, one has to smear on the four transverse directions to perform the transformation. The full NS5-brane background is characterized by the harmonic function *H* = *<sup>H</sup>*(*x*1, *x*2, *x*3, *x*<sup>4</sup>) in four Euclidean dimensions

$$
\triangle H(\mathbf{x}^1, \mathbf{x}^2, \mathbf{x}^3, \mathbf{x}^4) = h \, \delta^{(4)}(\mathbf{x}^1, \mathbf{x}^2, \mathbf{x}^3, \mathbf{x}^4). \tag{76}
$$

To have flat asymptotics at spatial infinity as *r* → ∞ one defines the harmonic function as

$$H = 1 + \frac{h}{r^2},\tag{77}$$

where *r*2 = (*x*<sup>1</sup>)<sup>2</sup> + (*x*<sup>2</sup>)<sup>2</sup> + (*x*<sup>3</sup>)<sup>2</sup> + (*x*<sup>4</sup>)2. To smear along say the direction *z* := *x*4 one considers harmonic function *H* = *<sup>H</sup>*(*x*1, *x*2, *x*<sup>3</sup>) which has apparent isometry along *z* and solves the equations of motion of supergravity with the same amount of flux. This procedure gives the following background of the smeared NS5-brane which we refer to as the H-monopole

$$\begin{aligned} ds^2 &= ds\_{056789}^2 + H ds\_{1234}^2, \\ B &= A \wedge dz, \\ e^{-2(\varphi - \varrho\_0)} &= H^{-1}. \end{aligned} \tag{78}$$

where *H* = 1 + *h*/*r* and the 1-form *A* = *Aidx<sup>i</sup>* is defined to be of a magnetic configuration

$$2\partial\_{[i}A\_{j]} = \epsilon\_{ijk}\partial\_{k}H,\tag{79}$$

hence the name "H-monopole". Performing T-duality along the compact direction *z* by simply applying the Buscher rules (11) one arrives at the background of KK-monopole

$$\begin{aligned} ds^2 &= ds\_{056789}^2 + H ds\_{123}^2 + H^{-1} (dx^4 + A)^2, \\ B &= 0. \end{aligned} \tag{80}$$

This background has vanishing B-field and the magnetic monopole configuration is given by the metric components *gzi* = *Ai*. In three transverse directions {*x*1, *x*2, *x*<sup>3</sup>} such background behaves as the gravitational magnetic monopole.

Already at this step the backgrounds start to ge<sup>t</sup> tricky. Indeed, the Gross-Perry monopole solution, which describes the {*t*, *x*1, *x*2, *x*3, *z*} part of the KK5 background, is topologically a Hopf fibration with the special cycle given by the *z* direction [60]. Moving around this cycle one observes periodicity in 4 *πh* and glues the background by a diffeomorphisms transformation. This non-trivial topological structure is reflected by non-vanishing geometric flux *fij z*, defined as structure constants of the algebra of local vielbeins

$$[e\_{a'}e\_b] = f\_{ab}{}^c e\_{c'} \tag{81}$$

where *a*, *b* = 1, 2, 3, *z* and the vielbein is defined in the usual lower-triangular gauge as

$$e\_i^a = H^{-\frac{1}{2}} \begin{bmatrix} H & 0 & 0 & 0 \\ 0 & H & 0 & 0 \\ 0 & 0 & H & 0 \\ A\_1 & A\_2 & A\_3 & 1 \end{bmatrix} \cdot \tag{82}$$

To T-dualize further, one has to smear one more direction, say *x*3, which brings us to the set of co-dimension-2 solutions. These are not well defined field configurations as one cannot satisfy proper conditions at space infinity asymptotics. Formally, the harmonic function in two dimensions is *<sup>H</sup>*(*ρ*) = *h*0 + *h* log *ρ*/*ρ*0, where *ρ*2 = (*x*<sup>1</sup>)<sup>2</sup> + (*x*<sup>2</sup>)2, and diverges both at the core and at the infinity. For that renormalization procedure introduces a dimensionful parameters *ρ*0, *h*0, which somehow run when one approaches the singularity points [17]. In what follows, we will drop these parameters for simplicity and always assume the harmonic function of the form

$$H = 1 + \tilde{h} \log \rho.\tag{83}$$

Performing T-duality transformation of the KK5-background smeared in such a way one arrives at the following background

$$\begin{aligned} ds^2 &= H(d\rho^2 + \rho^2 d\theta^2) + \frac{H}{H^2 + \tilde{h}^2 \theta^2} ds\_{34}^2 + ds\_{056789}^2, \\ B^{(2)} &= \frac{\tilde{h}\theta}{H^2 + \tilde{h}^2 \theta^2} dx^3 \wedge dx^4, \\ e^{-2(\varphi - \varphi\_0)} &= \frac{H}{H^2 + \tilde{h}^2 \theta^2} \end{aligned} \tag{84}$$

where {*ρ*, *θ*} are the polar coordinates in the transverse plane {*x*1, *<sup>x</sup>*<sup>2</sup>}. One immediately notices the non-trivial monodromy: when going around the brane *θ* → *θ* + 2*π* the background is glued by a T-duality transformation in the directions {*x*3, *<sup>x</sup>*<sup>4</sup>}. Such non-geometric configurations are believed to be sourced by exotic branes and are properly described on the language of T-folds [15,16,61]. In terms of the classification (28) the background (84) is sourced by the 522-brane. Now it is easy to see that the two "quadratic" directions in the mass formula correspond to the two special directions of the brane.

The monodromy gluing the torus {*x*3, *x*<sup>4</sup>} can be conveniently represented as a linear transformation of the corresponding generalized metric

$$\mathcal{H}(\boldsymbol{\theta}^{\prime} = \boldsymbol{\theta} + 2\pi) = \mathcal{O}^{\mathrm{tr}}\mathcal{H}(\boldsymbol{\theta})\mathcal{O},\tag{85}$$

where the matrix O encodes the non-geometric *β*-transform

$$\mathcal{O} = \begin{bmatrix} \mathbf{1\_2} & 0\\ \beta(\theta') & \mathbf{1\_2} \end{bmatrix} \tag{86}$$

with *β*(*θ*) = ˜ *hθ ∂*3 ∧ *∂*4. This suggests that it is more natural to use the *β*-frame of DFT [62] where the generalized metric parametrized by the space-time metric and a bi-vector field *βab* instead of the 2-form Kalb-Ramond field. In these variables the background becomes

$$\begin{split} ds^2 &= H(d\rho^2 + \rho^2 d\theta^2) + H^{-1} ds\_{34}^2 + ds\_{056789\prime}^2 \\ \beta &= \beta^{34} \frac{\partial}{\partial x^3} \wedge \frac{\partial}{\partial x^4} \,. \end{split} \tag{87}$$

This form naturally reflects the fact that the 522-brane electrically interacts with the bi-vector field as was shown explicitly in [20,63,64]. Indeed, the bi-vector *βab* is actually an 8-form taking values in two-vectors, which can be written in 10-dimensions as

$$\mathcal{B}\_{(8,2)} = \beta\_{\mu\_1,\dots,\mu\_8}{}^{\nu\_1 \nu\_2} \, dx^{\mu\_1} \wedge \dots \wedge dx^{\mu\_8} \, \partial\_{\nu\_1} \wedge \partial\_{\nu\_2 \nu} \tag{88}$$

with additional condition that all components with any of the upper indices repeating any of the lower indices vanish [58]. Hence, the corresponding Wess-Zumino term for the 522-brane has the following structure

$$S\_{WZ}^{5\_2^2} = h \int d^6 \xi \beta\_{a\_1...a\_612} \, ^{34} d\xi^{a\_1} \wedge \cdots \wedge d\xi^{a\_6} \, \_{\prime} \tag{89}$$

where *ξα* are the world-volume coordinates.

Turning to the *β*-frame allows defining the corresponding flux explicitly as *Qabc* = *<sup>∂</sup>aβbc*, which in 10 dimensions is represented by a (9, 2) mixed symmetry tensor. Smearing one more direction *x*2 one obtains linearly growing harmonic function, and T-dualising along *x*2 arrives at the s-called R-monopole, the 532-brane. This is a co-dimension-1 solution, a domain wall, which is even more peculiar. Non-vanishing R-flux *Rabc* = <sup>3</sup>*β<sup>d</sup>*[*a∂dβbc*] of the background reflects non-associativity of the closed string coordinates on such background [65,66].

One concludes that exotic brane backgrounds in the approach of the conventional supergravity are solutions fo equations of motion of co-dimension ≤ 2, i.e., domain strings, walls and space-time filling objects. Co-dimension-2 solutions are characterized by non-trivial monodromy around the core of the object, which glues the background by a T-duality transformation. In the T-duality covariant approach of DFT one expects to describe a whole duality orbit as a single field configuration. This is indeed possible and has been shown explicitly for the orbit of the NS5-brane in [67,68] and for backgrounds of exotic branes of M-theory in [69–71] (for review see [72]).

In [67,68] it has been shown that all branes, both geometric and exotic, of the T-duality orbit starting with the NS5-brane are just different faces of a single object called DFT monopole. This background is a solution of DFT equations of motion, characterized by the generalized metric written in the form of a quasi-interval *ds*<sup>2</sup> = H*MNd*X*Md*X*<sup>N</sup>* (note that this is not an invariant expression and cannot be understood as a definition for distance in the doubled space) and the invariant dilaton *d*

$$\begin{split} ds\_{\mathrm{DFT}}^2 &= H(1 + H^{-2}A^2)d\dot{z}^2 + H^{-1}d\dot{z}^2 + 2H^{-1}A\_i(d\dot{y}^i d\dot{z} - \delta^{ij}d\vec{y}\_j dz) \\ &+ H(\delta\_{\dot{\imath}\dot{\jmath}} + H^{-2}A\_i A\_j)d\dot{y}^i d\dot{y}^j + H^{-1}\delta^{ij}d\vec{y}\_i d\vec{y}\_j \\ &+ \eta\_{\mathrm{fr}}d\mathbf{x}^r d\mathbf{x}^s + \eta^{rs}d\vec{x}\_r d\vec{x}\_s, \\ e^{-2d} &= H e^{-2\rho y}. \end{split} \tag{90}$$

Here the harmonic function and the vector *Ai* are that of the H-monopole

$$\begin{aligned} H(y) &= 1 + \frac{h}{\sqrt{\delta\_{ij} y^i y^j}}, \\ 2\partial\_{[i} A\_{j]} &= \epsilon\_{ijk} \partial\_k H. \end{aligned} \tag{91}$$

Of crucial importance here is the understanding of the coordinate dependence of the fields. First, one notes that the section condition is satisfied since the fields depend only on (*y*1, *y*2, *y*<sup>3</sup>) and do not depend on their duals (*y*˜1, *y*˜2, *y*˜3). Second, the form of the solution as written above does not tell us which coordinates are geometric, i.e., used for measuring physical distances, and which are not. This is additional information which is given upon fixing position of the DFT monopole in the doubled space. Since nothing changes when replacing *x<sup>r</sup>* by *<sup>x</sup>*˜*r*, equivalently, when T-dualizing along the world-volume directions, one has five possible choices for geometric coordinates *xμ* listed in Table 3. Consider the Q-monopole, the 522 brane, for which the geometric set of transverse coordinates is (*z*˜, *y*˜1, *y*2, *<sup>y</sup>*<sup>3</sup>). This implies that the harmonic function, which is always of the form (91), depends on one dual coordinate that is *x*˜1 = *y*1 in this case. Hence, one has properly defined harmonic function with nice asymptotic behaviour, which however is allowed to depend on dual coordinates. Please note that instead of the H-monopole background one may start with the full NS5-brane background and the harmonic function

$$H(z, y) = 1 + \frac{h}{z^2 + \delta\_{ij} y^i y^j},\tag{92}$$

which would imply that a <sup>5</sup>*<sup>r</sup>*2-brane background is characterized by fields which depend on *r* dual coordinates. This number if also equal to the number of special circles.

The orbit starting at the NS5-brane was also considered in [67], where the background of the localised Kaluza-Klein monopole has been recovered from the DFT monopole. Such a background has been known before these results and was recovered from the usual Kaluza-Klein monopole background by taking into account backreaction from worldsheet instantons in the two-dimensional linear sigma model [73–76]. Similarly, worldsheet instanton corrections have been shown to localize the background of the 522 co-dimension-2 brane in dual space [77]. In Figure 4 relations between monopoles and their localized versions are shown schematically.

**Figure 4.** Systematics of backgrounds with H, geometric f and Q fluxes and their relations. Note that in [77] Q-monopole localised in both *X*<sup>8</sup> and *X*<sup>9</sup> has been constructed with both these directions being compact.


**Table 3.** Possible choices for orientation of DFT monopole in the doubled space.

#### *3.3. Deformations of Supergravity Backgrounds*

Backgrounds of DFT depending on a dual coordinate and the possibility to parametrize the generalized metric in terms of exotic degrees of freedom, such as the bi-vector *βμν*, open a window to investigate issues related to integrability of the 2-dimensional sigma model. It is known that kappa symmetry of the two-dimensional sigma model does not imply strictly equations of motion of 10-dimensional supergravity, but leads to a more general setup. This has been elaborated in a series of paper (see e.g., [78–81]) and is usually referred to as the generalized supergravity. In this theory the degree of freedom represented by the dilaton and its derivative is replaced by a set of vectors (*Xμ*, *Zμ*, *<sup>I</sup><sup>μ</sup>*), and the equations of motion of generalized supergravity drop to the conventional ones upon *<sup>X</sup>μ* = *∂μϕ*. The most amazing feature of this theory is that (at least some) solutions of its equations of motion can be obtained as integrable deformations of backgrounds of the conventional sugra.

Integrability of the two-dimensional sigma model on the *AdS*5 × S5 background has been shown in [82] by explicit construction of the corresponding Lax pair. Such sigma model gain the expected interpretation as Type II string theory living on the *AdS*5 × S5 background supported by non-vanishing flux of 5-form. In [83] a deformed version of the *AdS*5 × S5 sigma-model has been considered, which appeared to be also integrable for the so-called *η*-deformations. The *η*-deformed sigma model is very peculiar when understood as theory of string on a classical background. In this case, the corresponding background, which is commonly referred to as (*AdS*5 × <sup>S</sup><sup>5</sup>)*<sup>η</sup>*, does not satisfy equations of motion of 10-dimensional supergravity [84]. However, it still defines a proper background for string propagation due to Weyl invariance, and moreover the corresponding model is integrable. Such a deformed background was shown to satisfy equations of motion of generalized supergravity [81,85,86].

At the level of field theory, generalized supergravity can be obtained from the conventional Type II supergravity by considering equations of motion for background with one isometry direction *x*<sup>∗</sup> and with dilaton, linearly depending on the isometry direction *ϕ* = *ϕ*0 + *ax*<sup>∗</sup>. T-dualizing solutions along *x*<sup>∗</sup> by formally applying Buscher rules one obtains solutions of generalized supergravity. In Double Field Theory such procedure is reflected by just changing the role *x*<sup>∗</sup> as a geometric coordinate to become non-geometric, as has been shown explicitly in [86,87]. Alternatively, one is able to reproduce generalized supergravity equations of motion by generalized Scherk-Schwarz reduction of exceptional field theory with twist matrices depending on one dual coordinate. This was explicitly shown for the *E*6 ExFT in [88]; however, similar procedure can be performed for ExFTs with any U-duality group.

In addition to the nice incorporation of both generalized and normal supergravities in a single picture, DFT provides a simple and straightforward algorithm for generating deformed backgrounds from a given solution. This procedure has been mainly developed in [89–94] and is based on the so-called *β*-frame of DFT [95]. The basic idea of *B vs β* frame is that the generalized metric of DFT as an element of the coset space <sup>O</sup>(*d*, *d*)/O(*d*) × O(*d*) can be parametrized in terms of the fields *<sup>G</sup>μν*, *<sup>B</sup>μν* or alternatively *gμν*, *βμν*

$$\mathcal{H}\_{MN} = \begin{bmatrix} G\_{\mu\nu} - B\_{\mu\rho} B^{\rho}{}\_{\nu} & -B\_{\mu}{}^{\nu} \\ B^{\mu}{}\_{\nu} & G^{\mu\nu} \end{bmatrix} = \begin{bmatrix} g\_{\mu\nu} & -b\_{\mu}{}^{\nu} \\ b^{\mu}{}\_{\nu} & g^{\mu\nu} - \beta^{\mu\rho} \theta\_{\rho}{}^{\nu} \end{bmatrix}. \tag{93}$$

*Symmetry* **2019**, *11*, 993

This is equivalent to choosing the generalized vielbein to be of lower-triangular or upper-triangular form. In principle, one may keep both *<sup>B</sup>μν* and *βμν* fields in the metric; however this will duplicate gauge degrees of freedom and require either an additional constraint or change in the status of one of the fields. When substituted into the full action of DFT

$$\begin{split} S\_{\rm HHZ} &= \int d\mathbf{x} d\mathbf{\bar{x}} e^{-2d} \left( \frac{1}{8} \mathcal{H}^{MN} \partial\_M \mathcal{H}^{KL} \partial\_N \mathcal{H}\_{KL} - \frac{1}{2} \mathcal{H}^{KL} \partial\_L \mathcal{H}^{MN} \partial\_N \mathcal{H}\_{KM} - \\ & \qquad - 2 \partial\_M d \partial\_N \mathcal{H}^{MN} + 4 \mathcal{H}^{MN} \partial\_M d \partial\_N d \right) . \end{split} \tag{94}$$

the generalized metric in *β*-frame provides the action of the so-called *β*-supergravity. This theory has been initially developed to address non-geometric fluxes *Q* and *R*, which are more naturally written in terms of the bivector field, as in the previous subsection.

The matrix equality (93) can be written in terms of the fundamental fields in the following simple form

$$(G+B)^{-1} = \mathcal{g}^{-1} + \mathcal{\beta}.\tag{95}$$

In the above one immediately recognizes the open-closed string map [96] that relates backgrounds as seen by the closed string to that seen by the open string. In this interpretation the field *βμν* on the RHS is understood as the non-commutativity parameter, usually denoted Θ*μν*.

The relation (93) can be understood as a deformation of a solution given by the metric background *gμν* by a coordinate-dependent parameter *βμν* (see Figure 5). One starts with a background with vanishing *B*-field, parametrized only by metric degrees of freedom *gμν* and the dilaton Φ. Since *<sup>B</sup>μν* = 0, the corresponding generalized metric can be understood in either *B* or *β*-frame. Consider now a deformed generalized metric, which contains the metric *gμν* that solves conventional equations of motion, and the field *β*. Then, written in the *B*-frame this generalized metric encodes a solution of equations of motion of the conventional supergravity. Equivalently, this implies that the generalized metric solves equations of motion of DFT given the section constraint is satisfied. In the *β*-frame, given *gμν* is a solution, these equations of motion impose constraints on the bi-vector field *βμν* for the deformation to give a solution. In [93] these constraints have been obtained explicitly from equations of motion of *β*-supergravity of [97]. Given a general solution of these equations one is guaranteed to recover solutions of generalized supergravity after deformation, and in the special case, when *∂μβμν* = 0, these boil down to solutions of the conventional supergravity.

**Figure 5.** Relationships between the relevant theories and their solutions. *b*-frame refers to the standard supergravity (possibly generalised), while *β*-frame is the theory of [98]. Yang-Baxter deformation acts within usual supergravity, but we interpret it as a composition of the open/closed string map with a deformation by *βmn*. This leads to the constraints for *βmn* (essentially the CYBE) arising from supergravity field equations.

Of special interest are backgrounds with a set of isometries encoded by Killing vectors *ka* = *ka μ∂μ*, which form an algebra upon

$$[k\_a, k\_b] = f\_{ab}{}^c k\_c.\tag{96}$$

For these the deformation parameter can be chosen in the bi-Killing form

$$\mathcal{B}^{\mu\nu} = k\_a{}^{\mu}k\_b{}^{\nu}\tau^{ab}{}\_{\prime} \tag{97}$$

where *rab* is a constant antisymmetric matrix. Such deformed backgrounds satisfy equations of motion of supergavity if the *r*-matrix satisfied classical Yang-Baxter equation, which is the semi-classical limit of the quantum Yang-Baxter equation

$$r^{|a|b|}r^{c|d|}f\_{bd}{}^{c|} = 0.\tag{98}$$

A particular example of such deformation is given by the (*AdS*5 × <sup>S</sup><sup>5</sup>)*η*-background discussed above, which corresponds to deformations with *r*-matrix *rab fabc* = *Ic*. The non-vanishing vector *Ic* is a sign of the generalized supergravity equations of motion, and indeed *∂μβμν* = *ka ν Ia*.

The constraint imposed on couplings of the two-dimensional sigma-model by Weyl invariance has been uplifted to the full T-duality covariant sigma-model in [94,99] and proper local counter-terms have been constructed. The covariant approach implies a generalization of the bi-vector deformations described above into a full *β*-shift. This is a local transformation inside DFT, which can be applied to a solution with non-vanishing *H*-flux. The general procedure is as follows, one starts with a generalized metric H*MN* and a set of generalized Killing vectors *Ka M*, the deformation then is given by

$$\mathcal{H}'\_{\rm MN} = h\_M{}^K \mathcal{H}\_{\rm KL} h\_N{}^L \prime \quad h\_M{}^N = \delta\_M{}^N - 2\eta r^{ab} K\_{aM} K\_b{}^N \prime \tag{99}$$

where *η* is a constant parameter of deformation. The generalized Killing vector should satisfy the algebraic section constraint (cf. that of [64])

$$
\eta\_{MN} \mathbf{K}\_a^M \mathbf{K}\_b^N = 0.\tag{100}
$$

This ensures the matrix *hM N* to be a T-duality transformation, i.e., *h* ∈ <sup>O</sup>(*d*, *d*). In the usual duality frame one chooses *Ka M* = (*ka μ*, 0) and the above transformation reduces to a *β*-deformation.

#### **4. Conclusions and Discussion**

In this short review, we have briefly described the structure of exceptional field theories (ExFT's), which provide a (T)U-duality covariant approach to supergravity. These are based on symmetries of toroidally reduced supergravity; however are defined on a general background. From the point of view of ExFT the toroidal background is a maximally symmetric solution preserving all U-duality symmetries. In this sense the approach is similar to the embedding tensor technique, which is used to define gauge supergravity in a covariant and supersymmetry invariant form. Although any particular choice of gauging breaks certain amount of supersymmetry, the formalism itself is completely invariant. Similarly the U-duality covariant approach is transferred to dynamics of branes in both string and M-theory, whose construction has not been covered here.

In the text, we described construction of the field content of exceptional field theories from fields of dimensionally reduced 11-dimensional supergravity, and local and global symmetries of the theories. Various solutions of the section constraint giving Type IIA/B, 11D and lower-dimensional gauged supergravities have been discussed without going deep into technical details. For readers' convenience references for the original works are present.

As a formalism exceptional field theory has found essential number of application, some of which have been described in this review in more details. In particular, we have covered generalized twist reductions of ExFTs, which reproduce lower-dimensional gauged supergravities, description of non-geometric brane backgrounds and an algorithm for generating deformations of supergravity backgrounds based on frame change inside DFT. However, many fascinating applications of the DFT and ExFT formalisms have been left aside. Among these are non-abelian T-dualities in terms of Poisson-Lie transformations inside DFT [100,101]; generating supersymmetric vacua and consistent truncations of supergravity into lower dimensions [102–104] (for review see [105]); compactifications on non-geometric (Calabi-Yau) backgrounds and construction of cosmological models [54,55,106,107].

**Funding:** This work was funded by the Russian state gran<sup>t</sup> Goszadanie 3.9904.2017/8.9 and by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS".

**Conflicts of Interest:** The author declares no conflict of interest.
