Consistent sphere reductions of gravity to two dimensions

Consistent reductions of higher-dimensional (matter-coupled) gravity theories on spheres have been constructed and classified in an important paper by Cveti\v{c}, L\"u and Pope. We close a gap in the classification and study the case when the resulting lower-dimensional theory is two-dimensional. We construct the consistent reduction of Einstein-Maxwell-dilaton gravity on a $d$-sphere $S^d$ to two-dimensional dilaton-gravity coupled to a gauged sigma model with target space ${\rm SL}(d+1)/{\rm SO}(d+1)$. The truncation contains solutions of type AdS$_2\times \Sigma_d$ where the internal space $\Sigma_d$ is a deformed sphere. In particular, the construction includes the consistent truncation around the near-horizon geometry of the boosted Kerr string. In turn, we find that an AdS$_2\times S^d$ background with the round $S^d$ within a consistent truncation requires $d>3$ and an additional cosmological term in the higher-dimensional theory.


Introduction
Consistent truncations of higher-dimensional gravity theories have a long history nourished by the emergence of extra dimensions in supergravity and string theory.Explicitly, this is the question if a gravitational theory can be truncated to a finite set of fields whose dynamics is described by a lower-dimensional action, such that any solution of the lower-dimensional theory can be uplifted to a solution of the original, higher-dimensional theory.In terms of the infinite towers of Kaluza-Klein fluctuations around a given background, this corresponds to the truncation to a finite number of such fluctuations which is consistent at the full non-linear level.In general, non-linear products of the fields that are being retained will act as sources for the truncated fields, thereby rendering the truncation inconsistent.Consistent truncations are not low energy effective field theories since retained fields may have masses comparable to the truncated fields.Yet, they provide very powerful tools for the construction of higher-dimensional exact solutions as well as for the applicability of supergravity techniques for holographic dualities.
Unlike for toroidal reductions which retain precisely the singlets under the U(1) d isometry of the torus T d such that consistency follows from a simple group-theoretic argument, the question becomes much more involved for non-trivial internal spaces.In an important paper, Cvetič, Lü and Pope have classified and explicitly constructed the consistent reductions of (matter-coupled) gravity theories on spheres S d [1], that retain all the Yang-Mills fields of SO(d + 1), gauging the full isometry group of the sphere.A strong necessary condition for the existence of such a consistent truncation can be found from the toroidal reduction of the relevant higher-dimensional theory.Its global symmetry group G must accommodate an SO(d + 1) subgroup such that gauging of the latter describes the theory obtained from reduction on the sphere.In general, this requires some symmetry enhancement that only occurs for particular matter content and couplings of the higherdimensional theory.For example, a straightforward counting argument along these lines [1] shows that the reduction of D-dimensional gravity coupled to a single d-form field strength F (d) on the sphere S d can only be consistent for (D, d) ∈ { (11,7), (11,4), (10,5)} . (1.1) These are precisely the reductions, realized for D = 11 supergravity on S 7 [2][3][4][5], D = 11 supergravity on S 4 [6], and IIB supergravity on S 5 [7], respectively.They describe consistent truncations around AdS D−d × S d backgrounds to the maximal supergravity multiplet, and as such have played important roles in the AdS/CFT dualities.
Further possibilities for such consistent sphere reductions exist when the higher-dimensional theory carries an additional dilaton field, i.e. is described by a D-dimensional Lagrangian with a particular value for the constant a = a(D, d) .As shown in [1], the symmetry enhancement required for existence of a consistent reduction on S d arises for the four families of theories with (D, d) ∈ (D, 2), (D, 3), (D, D − 3), (D, D − 2) , ( where the latter two are the Hodge duals of the first two.Furthermore, in [1] the consistent reductions on S 2 , S 3 , S D−3 , corresponding to the first three families of (1.3) were explicitly constructed.The last case of (1.3), i.e. reduction of the theory (1.2) on a sphere S D−2 has been left aside, mostly because the resulting theory is a two-dimensional gravity theory in which many of the generic structures degenerate.For instance, the theory enjoys an additional global Weyl symmetry and cannot be cast into the canonical Einstein frame.Furthermore, reductions on T (D−2) lead to ungauged gravity theories in two dimensions, which come with an infinite-dimensional symmetry enhancement, generalising the affine Geroch group, originally discovered in the dimensional reduction of D = 4 general relativity [8].
In this paper, we complete the classification of [1] and explicitly construct the consistent truncation of the theory (1.2) on a sphere S D−2 .The resulting theories are two-dimensional dilaton gravity theories which may carry AdS 2 vacua that uplift to higher-dimensional AdS 2 × S d backgrounds.As such they may provide important tools to describe fluctuations around the near-horizon geometry of (near-)extremal black holes.This is especially relevant for the study of such black holes in the context of AdS 2 holography, see for instance [9][10][11][12][13][14].
We present the full non-linear reduction ansatz for the fields of (1.2) on a sphere S D−2 .The matter sector of the resulting two-dimensional theories is a gauged sigma model with target space SL(d + 1)/SO(d + 1) and a scalar potential.The gauge fields appear with a two-dimensional YM term.Consistency of the truncation requires that the constant a in (1.2) must take a particular value.As it turns out, for this value the D-dimensional theory itself can be obtained by circle reduction of pure gravity in (D + 1) dimensions.Accordingly, we also work out the uplift of the two-dimensional theory to (D + 1) dimensions.For D = 10, our result coincides with the pure gravity sector of the ansatz constructed in [15,16], that describes the consistent truncation of D = 11 supergravity on S 8 × S1 using affine exceptional field theory [17][18][19].
We construct a number of solutions of the two-dimensional theories, mostly restricting to solutions with constant scalars and dilaton.We find multi-parameter families of such solutions living in the truncation of the two-dimensional theory to singlets under the U(1) Cartan subgroup of SO(d + 1).They naturally generalize the solutions found in [20] for the case of S 8 .Interestingly, we find that all such solutions necessarily break SO(d + 1), implying that the corresponding higherdimensional AdS 2 ×S d backgrounds all involve deformations of the round S d sphere.Notably, these solutions include the D = 5 near horizon geometry of the boosted Kerr string [21,22].
Embedding of an AdS 2 × S d background with the round S d on the other hand requires the addition of a cosmological term to the D-dimensional theory (1.2), with the constant b tuned to a particular value.Still, this turns out to be possible only for D > 5.In contrast, the D = 4 and D = 5 theories admit dS 2 × S 2 and Mink 2 × S 3 backgrounds, respectively, within their consistent truncations.The rest of this paper is organized as follows.In section 2. we start by reviewing the results of [1] on consistent S 2 and S 3 reductions from D dimensions.We describe how to properly extrapolate the constructions to D = 4 and D = 5, respectively, such that the resulting theories are twodimensional.In section 3 we generalize the structure to arbitrary dimensions and construct the consistent truncation of the theory (1.2) on a sphere S D−2 .We describe the further uplift to pure gravity in (D + 1) dimensions and the inclusion of a cosmological term (1.4).In section 4, we study solutions of the two-dimensional theories and their uplift to D dimensions.We close with some conclusions in section 5 where we also discuss the symmetries underlying the presented constructions.
2 Consistent S 2 and S 3 reductions

Review of previous results
We start by reviewing the results of [1] on consistent S d reductions from D dimensions for d = 2, 3.The D-dimensional theories are of the type (1.2), i.e. an Einstein-Maxwell dilaton system for d = 2, and the bosonic string with Kalb-Ramond field and a dilaton for d = 3.
Let us first describe the S 2 case: starting from the Einstein-Maxwell dilaton system 1 in D dimensions, the consistent reduction ansatz on an internal S 2 is given by Here, the µ i are the embedding coordinates of the S 2 sphere the symmetric and positive definite matrix T ij carries the lower-dimensional scalar fields, and The inverse sphere radius g appears as a coupling constant for the SO(3) covariant derivatives with the associated Yang-Mills field strength given by It has been shown in [1] that plugging the reduction ansatz (2.2) into the field equations obtained from (2.1), all the dependence on the sphere coordinates consistently factors out and the equations reduce to (D − 2)-dimensional field equations which are obtained from variation of the (D − 2)dimensional Lagrangian where the matrix T ij is parametrized as Tij such that det Tij = 1.The computation of the D-dimensional field equations also exploits the explicit form of the Hodge dual of F(2) from (2.2) which is found to be given by e − 2(D−1) in D dimensions, the consistent reduction ansatz on an internal S 3 is given by [1] . (2.10) Here, the µ i are the embedding coordinates of the S 3 sphere while the quantities ∆, Y , and U and the covariant derivatives are defined as in (2.4) and (2.5) above.On top of the SO(4) field strength (2.6), the lower-dimensional theory carries a two-form gauge potential B (2) whose field strength is defined as (2.12) It has been shown in [1] that plugging the reduction ansatz (2.10) into the field equations obtained from (2.9), all the dependence on the sphere coordinates consistently factors out and the equations reduce to (D − 3)-dimensional field equations which are obtained from variation of the (D − 3)dimensional Lagrangian where the matrix T ij is parametrized as Tij such that det Tij = 1.The computation of the D-dimensional field equations also exploits the explicit form of the Hodge dual of F(3) from (2.10) which is found to be given by e

Reductions to two dimensions
We have in the previous section reviewed the results of [1] on the consistent truncations of Ddimensional theories (1.2) on S 2 and S 3 .Although derived for higher dimensions, in principle, the entire construction goes through even in the case when the resulting theory is two-dimensional, i.e. for D = 4 on S 2 and D = 5 on S 3 .In particular, none of the reduction ansaetze (2.2), (2.10), diverges at these values for D. However, the resulting Lagrangians (2.7), (2.13), when evaluated in two dimensions, give rise to equations of motion that kill all of their dynamical content as a consequence of the two-dimensional Einstein equations.Consequently, the two-dimensional case was left apart in [1].Another indication that the Lagrangians (2.7), (2.13), do not get along well with a two-dimensional space-time is the fact that the reduction of the higher-dimensional Einstein-Hilbert term in general yields the two-dimensional Einstein-Hilbert term only up to a dilatonic prefactor.While in generic dimensions this factor can be removed by a Weyl transformation of the metric, this is no longer the case in two dimensions.A generic reduction to two dimensions will thus produce dilaton gravity rather than pure two-dimensional gravity.
In short, while the results of the previous section are still valid for the truncation to a twodimensional theory, the resulting theory does not describe a dynamical subsector of the higherdimensional theory.What we show here is how the situation can be remedied by properly redefining the fields in the above structures, before extrapolating the construction to two dimensions.
Consider first the S 2 reduction of the theory (2.1).Starting in general dimension D, we may redefine the (D − 2)-dimensional fields appearing in the ansatz (2.2) as Even though this rescaling is clearly singular at D = 4, the resulting ansatz (2.2) still has a smooth limit D → 4. Namely, defining in order to extract the Y -dependence of the various quantities, the ansatz (2.2) takes the form where now ds 2 D−2 refers to the new metric gµν from (2.15).So far, this is merely a rewriting of the ansatz (2.2) in different variables, but upon taking the limit D → 4, we obtain the smooth limit of D-dimensional metric and dilaton to which differs from the naive D → 4 limit of (2.2) by the non-trivial powers of ρ.Furthermore, the expression for F(2) in (2.17) has no explicit Y or g µν dependence and retains its form for D → 4. Accordingly, the resulting Lagrangian (2.7) after rescaling (2.15) also yields a non-trivial smooth limit to D → 4, given by again differing from the naive D → 4 limit of (2.7) by the non-trivial powers of ρ.Again, R and ⋆ refer to the new metric gµν from (2.15).We may now forget about the singular rescaling (2.15), and directly prove that the regular reduction ansatz (2.18) defines a consistent truncation of the 4-dimensional Einstein-Maxell dilaton system (2.1) on S 2 .We give the details in section 3.1 below, where we discuss the general S D−2 truncation.The resulting two-dimensional theory is given by the Lagrangian (2.19).For later convenience, we also note that the dual field strength (2.8) for D = 4 and in terms of the fields (2.15) is found to be In terms of this dual field strength, the original D = 4 theory (2.1) can be equivalently rewritten as Similarly, we can extend the general S 3 reduction of the previous section to the particular case D = 5 in which the resulting theory becomes two-dimensional.Starting from the S 3 reduction of the theory (2.9) in general dimension D, we may redefine the (D − 3)-dimensional fields appearing in the ansatz (2.2) as This rescaling is singular at D = 5, but the resulting ansatz (2.10) still has a smooth limit D → 5. Defining in order to extract the Y -dependence of the various quantities, the ansatz (2.10) takes the form where now ds 2 D−3 refers to the new metric gµν from (2.22).The limit D → 5 is smooth and yields for the D = 5 metric and dilaton, which differs from the naive D → 5 limit of (2.10) by the nontrivial powers of ρ.The expression for F(3) in (2.24) retains its form for D → 5, except for the first term F (3) which disappears.The resulting Lagrangian (2.13) after rescaling (2.22) also yields a non-trivial smooth limit to D → 5, given by again differing from the naive D → 5 limit of (2.13) by the non-trivial powers of ρ.Again, we may now directly prove that the regular reduction ansatz (2.26) defines a consistent truncation of the 5-dimensional bosonic string (2.9) on S 3 , and we give the details below.The resulting twodimensional theory is given by the Lagrangian (2.27).For later convenience, we also note that the dual field strength (2.14) for D = 5 and in terms of the fields (2.22) is found to be In terms of this dual field strength, the original D = 5 theory (2.9) can be equivalently rewritten as (2.29)

Consistent S D−2 reduction
It is now straightforward to generalize the results of the last section in order to define the general consistent S D−2 reduction of (1.2), thereby establishing the fourth family of consistent truncations in (1.3).Our starting point is the Einstein-Maxwell dilaton system obtained by dualizing 2 the d-form field strength F(d) into an abelian 2-form field strength F(2) , This is the Lagrangian obtained from the circle reduction of pure gravity in (D + 1) dimensions.

Reduction ansatz and two-dimensional theory
Extrapolating from the previous findings, we can construct a consistent truncation of the Ddimensional theory (3.1) on the sphere S d = S D−2 by the following reduction ansatz The matrix Tij now is a symmetric positive definite (d + 1) × (d + 1) matrix of unit determinant, and the quantities are defined as above.The inverse sphere radius g appears as a coupling constant for the SO(d + 1) covariant derivatives just as in (2.5) above.The corresponding two-dimensional SO(d + 1) field strength F µν ij is given by (2.6).
It is straightforward, although lengthy, to substitute the ansatz (3.2) into the D-dimensional equations of motion obtained from (3.1), and to verify that these consistently truncate to the field equations of a two-dimensional theory.We find that these equations can be derived from the following two-dimensional Lagrangian which in particular matches the above results (2. 19) and (2.27) for S 2 and S 3 , respectively, as well as the structures found for the S 8 case in [16,20,23].This is a two-dimensional dilaton gravity coupled to a gauged SL(d + 1)/SO(d + 1) coset space sigma model with a potential.The two-dimensional Lagrangian (3.4) can be equivalently rewritten as upon introducing auxiliary scalar fields . The latter are related to the Tij by the (nonabelian) duality equations and can be eliminated by virtue of their field equations  4 After some lengthy algebra, one can then prove the D-dimensional field equations as a consequence of (3.8).For the computation one only needs to focus on the terms involving one or two derivatives along the external directions.These contributions can be shown to cancel out entirely by repeatedly making use of the Schouten identity (3.10) 3 The D-dimensional volume form is given by ωD ..i d+1 , with the totally antisymmetric tensor ϵi 1 i 2 ...i d+1 , and ⋆ω2 = 1.
4 For d = 1, the formula degenerates to e 2 φ ⋆ F Next, by a similar computation one proves the Bianchi identity for F(2) from (3.2), which requires the two-dimensional fields to obey the field equations obtained from (3.4).Finally, 5 we can check the dilaton equations of the D-dimensional theory (3.1) and show that after some lengthy computation and heavy use of (3.10), again they reduce to particular combinations of the two-dimensional field equations.A useful identity in this computation is the expression for the Hodge dual c.f. identity (25) in [1].We have thus established that the reduction ansatz (3.2) indeed represents a consistent truncation of the theory (3.1) to the two-dimensional theory (3.4).

Uplift to D + 1
As discussed above, the D-dimensional theory (3.1) which we have consistently truncated on S D−2 , is itself the S 1 reduction of pure gravity in (D + 1) dimensions.It is thus a natural question to study the further uplift of the reduction ansatz (3.2) to (D + 1) dimensions.This can be achieved by the standard Kaluza-Klein formula where z denotes the additional circle coordinate, and where we used d = D − 2. The uplift thus requires an explicit expression for the gauge potential defining the 2-form field strength as Indeed, we can integrate up F(2) from (3.2) to the following expression whose exterior derivative can be shown to satisfy (3.15) after using the two-dimensional field equations.This yields another check on the Bianchi identity (3.11).The uplift of (3.2) to (D + 1) dimensions is then given by combining this ansatz with (3.14) into It represents a consistent truncation of (D +1)-dimensional Einstein gravity to the two-dimensional theory (3.4). 6n [24], the (D + 1)-dimensional uplift of the system (3.1) was used to provide an elegant explanation for the consistent truncation on S 2 found in [1].From the (D + 1)-dimensional point of view, the S 2 reduction of (3.1) corresponds to a standard Scherk-Schwarz reduction [25] on SU (2) .In contrast, here the reduction ansatz (3.17) does not provide any immediate insight as to why this reduction is consistent, rather it appears to represent a non-trivial truncation of pure gravity by itself.We will come back to this in the conclusions.

Cosmological term
Let us briefly study if the consistent truncation (3.2) is compatible with the presence of a 'cosmological term' in D dimensions.We find that the previous reduction ansatz still gives rise to a consistent truncation, with all internal coordinates factoring out from the D-dimensional field equations, if the dilaton power in (3.18) is given by For D = 4 and D = 5 this indeed coincides with the values extrapolated from the results of [1].
The effect of the cosmological term to the two-dimensional theory (3.4) is an additional term in the scalar potential given by It is interesting to note that the D-dimensional cosmological term (3.18) with dilaton power (3.19) is not compatible with the further uplift of the theory to (D + 1) dimensions, i.e. it does not arise by the S 1 reduction of a cosmological constant in (D + 1) dimensions.This is an indication of the fact that the non-abelian SO(d + 1) gauge group of the two-dimensional theory is not embedded into the geometric SL(d + 1) symmetry, arising in the toroidal reduction of (D + 1)-dimensional gravity.We will come back to this in the conclusions.

Solutions
In this section, we construct some solutions of the two-dimensional theory (3.4) and work out their uplift to D dimensions.In particular, we explore two-dimensional AdS 2 solutions, uplifting to higher-dimensional AdS 2 × Σ geometries.

Field equations
In order to construct solutions of the two-dimensional theory, let us first spell out the field equations derived from (3.5).In addition to the first-order equations (3.7) and (3.6) for the field strength and the auxiliary scalars, respectively, the scalar fields Tij satisfy the second order field equations Here, double brackets ((. . .)) refer to traceless symmetrization of indices.Furthermore, the Einstein and dilaton equations for (3.5) take the form In the following, we are going to construct particular solution to these equations.

SO(d + 1) invariant solution
Let us first consider solutions that preserve the entire SO(d + 1) symmetry of the theory.SO(d + 1) invariance requires such that the only non-trivial fields in the theory are the dilaton ρ and the two-dimensional metric.
With the domain-wall ansatz the Einstein field equations imply that Upon inserting this into (4.2),all remaining equations reduce to a single differential equation for the dilaton ρ(r) In particular, this shows that SO(d + 1) invariance is not compatible with a constant dilaton.The general solution of equation (4.6) carries one integration constant α (apart from the trivial shift freedom r → r + c), and can be given implicitly in terms of hypergeometric functions as In the limit α → ∞, we find the particular solution Using the reduction formulae (3.2) (and setting A 0 = 0), we find the D-dimensional uplift of this solution as For d = 8, this reproduces the half-supersymmetric domain wall solution of [26,27], corresponding to the ten-dimensional D0-brane near-horizon geometry.

AdS 2 solutions with constant scalars and dilaton
Here, we will search for solutions in which all scalars and the dilaton are constant, such that the two-dimensional metric becomes AdS 2 .In order keep things simple, we restrict to the Cartan truncation, i.e. the further consistent truncation of (3.5) to singlets under the Cartan U(1) subgroup of the SO(d + 1) gauge group.This truncation keeps only [ d+1 2 ] vector fields and [ d 2 ] scalar fields among the Tij , together with the dilaton ρ and the two-dimensional metric.By construction, all the retained fields are neutral under the remaining U(1) [ d+1 2 ] gauge group.It is technically useful to distinguish the cases of even and odd d, although the form of the resulting solutions is very similar.

d = 2k + 1
Let us parametrize the U(1) k+1 singlets within the scalar matrix Tij as in terms of k scalars h a > 0, a = 1, . . ., k, such that det Tij = 1 .Accordingly, we parametrize the matrix Y ij as in terms of k + 1 scalars y α , and similarly for the field strengths F µν ij .Plugging all this into the field equations (4.1), (4.2), shows that all fields can be determined as a function of the free parameters h a and ρ as In particular, the two-dimensional metric is AdS 2 .

d = 2k
Similar to (4.10), in this case we parametrize the U(1) k singlets within the scalar matrix Tij as in terms of k scalars h a > 0, such that det Tij = 1 .Accordingly, we parametrize the matrix Y ij as All other fields can then be determined as a function of the remaining free parameters h a and ρ as Again, the two-dimensional metric is AdS 2 .For d = 8, this reproduces the solution (4.16) reproduces the solutions of [20].For d = 2, the condition (4.15) implies that h 1 = 1 2 , and the only free parameter in (4.16) is the constant dilaton ρ, which we may absorb into a rescaling of the fields.After setting g = ℓ −1 and some further rescaling of fields, the four-dimensional solution takes the form with ∆ 0 = (1 + cos 2 θ), and we have parametrized µ = {sin θ cos ϕ, sin θ sin ϕ, cos θ}.The metric and volume form of AdS 2 are denoted by ds 2 AdS 2 and ω2 , respectively, with AdS radius ℓ.Moreover, we have introduced Using (3.17), we may further uplift this solution to a solution of pure gravity in D = 5 dimensions, given by the metric Using an explicit parametrization of the AdS 2 metric in coordinates {v, r}, as well as for the 1-form A, the metric (4.19) takes the form (4.20) where we have defined The metric (4.20), (4.21) has isometry group SO(2, 1)×U(1) 2 , with the first factor realized on AdS 2 and the two abelian factors realized by shifts of the periodic coordinates ϕ and z.Up to redefinition of ϕ and z, this precisely reproduces the near-horizon of the boosted Kerr string, c.f. [21,22,28,29].The two-dimensional theory (3.5) for d = 2 thus captures a consistent truncation around this near horizon geometry.

(A)dS 2 × S d solutions of higher dimensional theory with cosmological term
We have seen in section 3.3 that the consistent truncation supports the presence of a 'cosmological term' (3.18) in D dimensions, which changes the scalar potential of the two-dimensional theory.We may thus for this case reexamine the existence of SO(d+1) invariant solutions (4.3).Evaluating the new scalar potential in presence of (3.20), we find that the theory admits a solution with constant dilaton ρ = ρ 0 , provided the coefficient of the cosmological term is given by

.22)
In this case, the two-dimensional curvature scalar is determined from (4.2) as

Conclusions
In this paper, we have completed the classification of [1], and worked out the consistent truncation of D-dimensional Kaluza-Klein gravity (3.1) on an S D−2 sphere to two dimensions.We have given the two-dimensional Lagrangian and explicitly constructed several families of solutions as well as their uplift to D dimensions.In particular, we have identified within the consistent truncations several solutions with AdS 2 geometry that uplift to different AdS 2 × Σ geometries, and notably the near-horizon geometry of the boosted Kerr string.This construction realizes the fourth and last of the families listed in (1.3), identified in [1] as potentially consistent sphere truncations.As discussed in the introduction, the existence of these truncations requires the embedding of the sphere isometry group into the global symmetry group of the toroidally compactified theory.Generically, the toroidal reduction of (1.2) on T d admits an R × GL(d) global symmetry,7 however this symmetry is enhanced at particular values of a, which is realized for the four families of (1.3).While this symmetry enhancement is only a necessary condition for consistency of the sphere truncations, their existence can be proven by explicit computation as has been done in [1] and the present paper.With hindsight though, as has become apparent in recent years, all the consistent sphere truncations corresponding to (1.1) and (1.3) owe their existence to some underlying symmetry structure in Riemannian, generalized, and exceptional geometry.
For the S 2 reductions, i.e. the first family of (1.3), it was shown in [24] that the consistent truncation of [1] becomes more transparent after further uplift of the D-dimensional theory (1.2) to pure gravity in (D + 1) dimensions (for a given as in (2.1)).In terms of this higher-dimensional theory, the truncation amounts to a standard Scherk-Schwarz reduction [25] on the group manifold SU(2).In particular, this explains the symmetry enhancement of the toroidal reduction of (1.2).The SO(3) isometry group of the sphere reduction is then naturally embedded into the enlarged symmetry group.For the S 3 reductions, i.e. the second family of (1.3), the consistent truncation requires the symmetry enhancement [30] R of the toroidal reduction of (1.2) (for a given as in (2.9)), such that the SO(4) isometry group of the sphere can be embedded into the enlarged symmetry group.The existence of the consistent truncation can be attributed to the generalized parallelizability of this background within the double field theory formulation of (1.2) [31][32][33].For the third family of (1.3), i.e. the truncation on S D−3 = S d , the further symmetry enhancement after toroidal reduction to three dimensions allows to embedd the SO(d + 1) isometry group of the sphere into the enlarged symmetry group This consistent truncation then has a natural explanation in the framework of the enhanced double field theory of [37].
In the same spirit, the existence of the consistent truncation on S D−2 constructed in this paper, allows for a natural explanation in the framework of the affine exceptional field theory of [17][18][19], as discussed in detail in [15,16,38].Its starting point is the symmetry enhancement after toroidal reduction of (1.2) to two dimensions (for a given as in (3.1)) according to [8,[39][40][41] R × GL(d) → GL(d + 1) → R ⋉ SL(d + 1) , (5.4) where SL(d + 1) denotes the affine extension of SL(d + 1).We have observed in section 3.3 that the addition of a cosmological term is compatible with the consistent truncation but obstructs the uplift of (1.2) to D + 1 dimensions.This is a manifestation of the fact that the SO(d + 1) isometry group of the sphere is not embedded into the intermediate GL(d + 1).This was already noticed in [15,16], to which we refer for more details.An immediate application of consistent truncations is the construction of higher-dimensional solutions.By construction, any solution of the lower-dimensional theory uplifts into a solution of the higher-dimensional theory.In particular, solutions with constant scalar fields, that take a simple form in the lower-dimensional theory may give rise to higher-dimensional backgrounds with complicated internal geometry, which has been exploited in many instances in the past.As an illustration, we have constructed a few solutions of the two-dimensional theory (3.4) together with their higher-dimensional uplift, but it would certainly be very interesting to generalize this to a more systematic and exhaustive construction of solutions.Within the Cartan truncation, it should be straightforward to work out the general rotating brane solutions as in [20,42,43] and study their thermodynamic properties using Sen's entropy function formalism [44].Going beyond the Cartan truncation of the scalar sector will require to deal with non-trivial non-abelian gauge fields and may lead to entirely new classes of higher-dimensional solutions.
As another application of consistent truncations, these provide valuable tools in the context of holography, since they allow to perform supergravity calculations directly within the lowerdimensional theory.E.g. this offers an immediate path to the computation of conformal dimensions and correlation functions of the operators dual to the fields of the consistent truncation.We have shown that the two-dimensional Lagrangian (3.4) carries a solution that uplifts to the near-horizon geometry of the boosted Kerr string (4.20).It will be very interesting to systematicaly analyze within this theory the perturbations around this background in the context of the Kerr/CFT correspondence [45], see e.g.[46] for a related study.More recently, it has been shown that consistent truncations together with the underlying exceptional geometry may even allow to access the full Kaluza-Klein spectrum, i.e. the infinite towers of fluctuations, around a given background [47,48].
It would be highly interesting to develop similar technology around the backgrounds considered here, and beyond, with potential applications to compactifications of maximal supergravity, see e.g.[49][50][51][52].We hope to come back to this in the future.

. 8 )
Throughout, ⋆ refers to the Hodge star in D dimensions, while ⋆ refers to the Hodge star in (D − 2) dimensions.The volume form of the (D − 2) dimensional metric is denoted by ω D−2 .Consistent truncations on S 3 are constructed in an analogous way.Starting from the Lagrangian of the bosonic string

. 7 ) 3 e 2
In order to prove consistency of the ansatz (3.2), it is useful to first work out the Hodge dual of the D-dimensional 2-form field strength F(2) from (3.2) as

. 14 )
in terms of k scalars y a .Plugging all this into the field equations (4.1), (4.2), yields the condition

4. 3 .3 d = 2
Using the reduction formulae (3.2), we can construct the D-dimensional uplift of the above solutions.They describe D-dimensional AdS 2 × Σ d backgrounds where Σ d is a deformed S d -sphere preserving only U(1) [ d+1 2 ] ⊂ SO(d + 1) of the isometries of the round sphere.Rather than going through the general case, let us illustrate the uplift for the d = 2 case, i.e. uplift the solution (4.16) to D = 4 dimensions.

. 23 )
For d ≥ 4, i.e.D ≥ 6, we thus find a geometry AdS 2 × S d with the round sphere S d , if the D-dimensional theory carries a cosmological term with positive m 2 .In contrast, for d = 2, the consistent truncation supports a dS 2 × S 2 solution.For d = 3 (i.e. a reduction from D = 5 dimensions on S 3 ), the theory admits a solution of type Mink 2 × S 3 , living within the consistent truncation.