From Exceptional Field Theory to Heterotic Double Field Theory via K3
aa r X i v : . [ h e p - t h ] J un LMU-ASC 62/16
From Exceptional Field Theory to Heterotic Double FieldTheory via K3
Emanuel Malek
Arnold Sommerfeld Center for Theoretical Physics, Department f¨ur Physik,Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, 80333 M¨unchen, Germany
Abstract
In this paper we show how to obtain heterotic double field theory from exceptionalfield theory by breaking half of the supersymmetry. We focus on the SL(5) exceptionalfield theory and show that when the extended space contains a generalised SU(2)-structuremanifold one can define a reduction to obtain the heterotic SO(3 , n ) double field theory. Inthis picture, the reduction on the SU(2)-structure breaks half of the supersymmetry of theexceptional field theory and the gauge group of the heterotic double field theory is given bythe embedding tensor of the reduction used. Finally, we study the example of a consistenttruncation of M-theory on K3 and recover the duality with the heterotic string on T . Thissuggests that the extended space can be made sense of even in the case of non-toroidalcompactifications. ontents N = 2 truncations 5 N = 2 truncations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.1 Truncation Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Consistency conditions and embedding tensor . . . . . . . . . . . . . . . . 10 , n ) section condition . . . . . . . . . . . . . . . 134.4 Heterotic generalised Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Introduction
Exceptional field theory [1–3] is an E d ( d ) -manifest extension of supergravity which has beenshown to include 11-dimensional and IIB SUGRA in one unified formalism. The starting pointfor exceptional field theory (EFT), just as for generalised geometry [4,5], is a Kaluza-Klein split of11-dimensional SUGRA. The bosonic and fermionic degrees of freedom then form representationsof the exceptional groups and their maximal compact subgroups, respectively. In EFT, one alsoextends the coordinates to form a representation of the exceptional groups.When one considers the E d ( d ) EFT on a d -torus, the interpretation of the exceptional groupand the extra coordinates becomes very clear. The exceptional group, or rather its integerpart E d ( d ) ( Z ), represents the U-duality group, and the extended coordinates are the Fourierduals of momentum- and wrapping-modes of branes. However, on more general backgroundsthe interpretation of the extended coordinate space is much less clear. However, in the caseof double field theory (DFT) [6, 7], which is an O( d, d )-manifest extension of type II SUGRAfollowing in the footsteps of earlier work [8–13], one can interpret the doubled space as arisingfrom the independent zero-modes of the left- and right-movers of the string, and one might expecta similar picture in the case of EFT.With this in mind, we here wish to study exceptional field theories on backgrounds withnon-trivial structure group. To be concrete we work with the SL(5) EFT which has a seven-dimensional “external” space, and a 10-dimensional “extended internal” space. In this caseone can consider a background which has generalised SU(2)-structure [14, 15] in which casethe background breaks half of the supersymmetry of the exceptional field theory. A particularexample of such a background would be a K3 surface.In [14] the distinction is drawn between the linear symmetry group of a theory and theduality group of the truncation on a particular background. The linear symmetry group of atheory determines what representations its field content transforms under. For example the linearsymmetry group is GL( d ) in the case of d -dimensional general relativity, or E d ( d ) in the case ofexceptional generalised geometry and EFT, because of the inclusion of p -form field strengths.On the other hand, the duality group should here be understood as the symmetry groupacting on the moduli space of a truncation of the theory on a particular background. It is thisgroup which becomes the global symmetry group of the lower-dimensional gauged SUGRA and inprinciple this is different from E d ( d ) , and even much larger. In particular, for seven-dimensionalconsistent N = 2 truncations of EFT, [14] shows that the duality group becomes O(3 , n ) where n = 3 in general. When the background is generalised parallelisable, the linear symmetry andduality groups coincide explaining the emergence of the E d ( d ) groups as the global symmetrygroup for maximally supersymmetric consistent truncations. We will show that the extendedcoordinate space can be understood to enhance in a similar fashion with the duality group.In particular, we use the technology of [14] to show that exceptional field theory can bereduced to the heterotic double field theory [10, 11, 16] when the extended space contains ageneralised SU(2)-structure manifold. The generalised SU(2)-structure breaks half of the super-2ymmetry and the embedding tensor of the particular SU(2)-structure reduction that is useddefines the gauging of the heterotic double field theory. This is reminiscent of the procedureused in [17] to reduce exceptional field theory to massive IIA supergravity.From our work a picture emerges for the role of the extended coordinates of EFT on suchSU(2)-structure manifolds. A twisted version of the extended coordinates can be used to definethe ( n + 3)-dimensional extended space of the O(3 , n ) heterotic double field theory. Furthermore,the generalised Lie derivative of the SL(5) EFT reduces to the O(3 , n ) heterotic DFT, with thegauging determined by the reduction on the SU(2)-structure space. Indeed, the entire actionreduces to that of the heterotic O(3 , n ) DFT in the so-called “frame formalism” [16, 18].We also use these results to show how the duality between M-theory on K3 and the heteroticstring on T emerges in exceptional field theory. The K3 surface depends on four coordinates,which when chosen as part of the section, become the M-theory coordinates. On the other hand,if the section is chosen to exclude these four coordinates, we obtain a truncation of the abelianO(3 ,
19) heterotic DFT on T . Thus, a duality here corresponds to a change of section, in thesame way that a conventional U-duality does, as has been advocated in [19, 20]. In the languageof [21], this corresponds to a choice of polarisation.We begin with a short review of the SL(5) EFT in section 2 and a summary of the relevantfindings of [14] in section 3. In section 4 we then show how to perform the reduction Ansatz thatgives rise to the heterotic DFT. We also discuss how ( n + 3)-dimensional extended space emergesand how the SL(5) generalised Lie derivative reduces to the heterotic one. In section 5 we showthat the SL(5) EFT action reduces to that of the heterotic DFT in the frame formalism [18] andwith a Kaluza-Klein split [22] with the seven external dimensions. Finally, we discuss in section6 how the duality between M-theory on K3 and the heterotic string on T arises in exceptionalfield theory before concluding in section 7. Here we will give a very brief overview of the main ingredients of the SL(5) exceptional fieldtheory which we will require in the remaining discussion. We will introduce further conceptswhere they are needed along the way in the rest of the paper. We refer the reader to thereviews [23–25] and the papers [2, 3, 26] for more details.The SL(5) EFT can be viewed as a reformulation of 11-dimensional supergravity which makesthe linear symmetry group SL(5) manifest. Thus, the starting point is 11-dimensional supergrav-ity in a 7+4 split. Let us use x µ , µ = 1 , . . . ,
7, as coordinates for the “external” 7-d space andlabel y ¯ i , ¯ i = 1 , . . . , Y ab , forming the antisymmetric representation of SL(5), where we use a, b = 1 , . . . , M ab ∈ SL(5) / USp(4) . (2.1)Similarly all bosonic objects with one leg in the external space can be combined into 10 vectorfields A µab , those with two external legs can be combined into five two-forms B µν,a , etc.The local symmetries of 11-dimensional supergravity, i.e. diffeomorphisms and p -form trans-formations, also combine into a SL(5) action, generated by the so-called generalised Lie derivative.For a tensor in the SL(5) fundamental representation V a of weight λ this takes the form [26–28] L Λ V a = 12 Λ bc ∂ bc V a − V b ∂ bc Λ ac + 15 V a ∂ bc Λ bc + λ V a ∂ bc Λ bc . (2.2)For consistency the algebra of generalised diffeomorphisms must close, i.e.[ L Λ , L Λ ] V a = L [Λ , Λ ] D V a . (2.3)Here the D -bracket just represents the action of a generalised Lie derivative,[Λ , Λ ] abD = L Λ Λ ab . (2.4)In order for (2.3) to hold one needs to impose the so-called section condition [26, 27] ∂ [ ab f ∂ cd ] g = 0 , ∂ [ ab ∂ cd ] f = 0 , (2.5)where f and g denote any two objects of the SL(5) EFT. There are two inequivalent solutions tothe section condition, one corresponding to 11-dimensional SUGRA and the other correspondsto type IIB. Upon using a solution of the section condition, the generalised Lie derivative (2.2)generates the p -form gauge transformation and diffeomorphisms of the corresponding SUGRA.Similarly, the EFT action reduces to that of 11-dimensional or IIB SUGRA [3, 29, 30], uponimposing a solution of the section condition.In exceptional field theory, and also double field theory, one can then interpret the choice ofsection as a duality transformation. In particular, the duality between strings and waves [19],branes and monopoles [20], and their non-geometric counterparts [31], can be seen to arise thisway. It has also been suggested that the M-theory / F-theory duality can be interpreted thisway [32]. Our work here also suggests that the heterotic / M-theory duality can also be seen asan exchange of solutions of the section condition.4 Summary of consistent N = 2 truncations SL(5)
EFT
In [14] it was shown how to construct seven-dimensional half-maximal consistent truncations ofthe SL(5) EFT. We will use this technology here to perform a half-maximally supersymmetricreduction of SL(5) EFT which yields the O( n,
3) heterotic double field theory. Let us begin byreviewing the relevant results of [14].A generalised SU(2)-structure is defined by the nowhere vanishing SL(5) tensors( κ, A a , A a , B u,ab ) , (3.1)where a = 1 , . . . , u = 1 , . . . , R and κ isa tensor density under generalised diffeomorphisms, which can be identified with the determinantof the external metric κ = | e | / . These tensors are subject to the compatibility conditions A a A a = 12 , B u,ab A b = 0 , B u,ab B v,cd ǫ abcde = 4 √ A e δ uv . (3.2)In [14] it was shown that any set of such tensors imply the existence of two nowhere-vanishingspinors and hence a truncation on such a background gives a half-maximally supersymmetrictheory. Because these tensors define a SU(2)-structure group and SU(2) ⊂ USp(4) they alsoimplicitly define a generalised metric.Furthermore, in [14], it was shown how to rewrite the entire EFT in terms of the SU(2)-structure ( κ, A a , A a , B u,ab ) instead of the generalised metric M ab . These can be thought ofas the exceptional analogue of the (almost) K¨ahler and (almost) complex structure of ordinarySU(2)-structure manifolds. To rewrite the action one introduces a generalised SU(2)-connectionwhose intrinsic torsion can be used to rewrite the generalised Ricci scalar of EFT, as well as theSUSY variations. The intrinsic torsion transforms in the following representations of SU(2) S × SU(2) R ⊂ SL(5): W int = 2 · ( , ) ⊕ ( , ) ⊕ · ( , ) ⊕ ( , ) ⊕ · ( , ) ⊕ ( , ) . (3.3)We refer readers who are interested in the definition of intrinsic torsion to [15, 33] as well as forthis particular case [14].In order to write the intrinsic torsion explicitly, we will make use of V uab = ǫ abcde B u,cd A e , and ˜ V uab = κǫ abcde B u,cd A e , (3.4)where ˜ V uab has weight and is thus a generalised vector. In addition we will need projectorsonto the ( , ) ⊂ , ( , ) ⊂ as SL(5) → SU(2) S × SU(2) R as well as a projector onto the5 , ) ⊂ ( , ) × ( , ) of SU(2) S × SU(2) R . These are given by P ab = δ ab − A a A b ,P abcd = (cid:18) δ abcd − √ B u,ab V ucd + 4 A [ a A [ c δ b ] d ] (cid:19) ,P au,bv = δ ab δ uv + √ B uac V vcb . (3.5)We can now write down the irreducible components of the intrinsic torsion. Singlets S = A a ∂ ab A b ,T = 112 κ ǫ uvw V u,cd L ˜ V v B wcd . (3.6)( , ) T u = − κ A a L ˜ V u (cid:0) A a κ − (cid:1) ,S u = 2 κ − L ˜ V u κ . (3.7)( , ) T ab = 112 κ P abcd L ˜ V u B ucd = 112 κ (cid:18) L ˜ V u B uab − √ B vab V vcd L ˜ V u B ucd + 4 A c A [ a L ˜ V u B ub ] c (cid:19) . (3.8)( , ) T uab = 112 κ ǫ uvw P abcd L ˜ V v B w,cd = 112 κ ǫ uvw (cid:18) L ˜ V v B w,ab − √ B xab V xcd L ˜ V v B w,cd + 4 A c A [ a L ˜ V v B | w | ,b ] c (cid:19) . (3.9)( , ) S a = 1 κ ∂ ab (cid:0) A b κ (cid:1) − A a A b ∂ bc A c ,T a = 112 κ ǫ uvw B u,ab V vbc L ˜ V w A c ,U a = 1 κ B u,ab L ˜ V u A b . (3.10)( , ) T ua = 1 κ P au,bv ǫ vwx B w,bc L ˜ V x A c . (3.11)6he generalised Ricci scalar R of the EFT is then given by R = 8 S − T − √ ST − T u T u + T u S u − S u S u − √ ǫ abcde T ab T cd A e − √ ǫ abcde T uab T u,cd A e − √ M ab S a S b − M ab S a T b + 83 M ab U a S b , (3.12)which is to be thought of as the half-maximal analogue to the flux formulation of DFT andEFT [34–40]. The EFT potential, which is defined as all the terms in the EFT action with onlyderivatives along Y ab , is in turn given by V = − R + 12 V uab V u,cd ˜ ∇ ab g µν ˜ ∇ cd g µν , (3.13)where ˜ ∇ ab is the SU(2)-connection. Because g µν is a SL(5) density of weight , the SU(2)-connection acts as ˜ ∇ ab g µν = | e | / ∂ ab (cid:16) g µν | e | − / (cid:17) . (3.14)Finally, the kinetic terms of the EFT action can also be written in terms of A a , A a and B u,ab instead of the generalise metric M ab . In [14] it was shown that they are given by L kin = 12 √ g µν ( D µ B u,ab D ν B ucd ) ǫ abcde A e − g µν D µ A a D ν A a + 18 F µν ab F µν cd (cid:0) B u,ab B ucd − B u [ ab B ucd ] (cid:1) − H µνρ,a H µνρb A a A b , (3.15)up to terms which vanish in a N = 2 theory.The full EFT action is then given by S = Z d Y d x | e | ( L EH + L kin − V ) + S top , (3.16)where L EH is the 7-dimensional external Einstein-Hilbert term and S top is the topological termof the EFT action [3, 41, 42] which require no further modification. N = 2 truncations As argued in [14], the truncation must not keep any doublets of the SU(2)-structure group inorder to yield an honest seven-dimensional N = 2 theory. In particular, a nowhere-vanishingdoublet of SU(2) S would imply that the structure group is actually trivial and there is underlying N = 4 supersymmetry. By decomposing SL(5) → SU(2) S × SU(2) R one finds that after removingthe SU(2) S doublets all remaining fields organise themselves into triplets and singlets of SU(2) S and SU(2) R .As a result, one can define a N = 2 truncation of the SL(5) theory by expanding all fieldsin terms of a basis of sections of the ( , )-, ( , )- and ( , )-bundles of SU(2) S × SU(2) R .7ecause SU(2) R is trivially fibred there can be only three sections of SU(2) R while SU(2) S isnon-trivially fibred and hence one can use n sections of the ( , ). n determines the number ofvector multiplets in the seven-dimensional half-maximal gauged SUGRA. We label these sectionsas ρ ( Y ) , n a ( Y ) , n a ( Y ) , ω M,ab ( Y ) , (3.17)where we have also introduced a SL(5) density ρ ( Y ). Here M = 1 , . . . , n + 3 labels the sectionsof the ( , )- and ( , )-bundles. These sections thus satisfy ω M,ab n b = 0 , (3.18)and we further normalise them according to n a n a = 1 , ω M,ab ω N,cd ǫ abcde = 4 η MN n e . (3.19)Here η MN has signature (3 , n ) reflecting the number of sections of the ( , )- and ( , )-bundlesused.It is useful to also introduce ω M ab = ǫ abcde ω M,cd n e , (3.20)which satisfy ω M ab ω N,ab = 4 η MN , ω M ab n b = 0 , (3.21)as a result of (3.19) and (3.18). Further useful identities are ω ( M cd ω N ) ca = η MN (cid:0) δ ab − n a n b (cid:1) ,ω M,ab ǫ abcde = 3 ω M [ cd n e ] ,ω M ab ǫ abcde = 12 ω M [ cd n e ] ,ω M ab ω N cd ǫ abcde = 16 η MN n e . (3.22)We will also often make use of the generalised vector˜ ω M ab = ρω M ab , (3.23)which has weight and thus can be used as a generator of generalised diffeomorphisms. One can now perform a truncation Ansatz by expanding all the fields of the SL(5) EFT in termsof ρ , n a , n a and ω M,ab . We will label the truncation Ansatz by the brackets hi . For the scalar8elds κ , A a , A a and B u,ab it is given by h κ i ( x, Y ) = | ¯ e | / ( x ) e − d ( x ) / ρ ( Y ) , h A a i ( x, Y ) = 1 √ e − d ( x ) / n a ( Y ) , h A a i ( x, Y ) = 1 √ e d ( x ) / n a ( Y ) , h B u,ab i ( x, Y ) = e − d ( x ) / b u,M ( x ) ω M ab ( Y ) , (3.24)and hence h V uab i = 1 √ e d ( x ) / b u,M ( x ) ω M,ab ( Y ) . (3.25)In order to satisfy the compatibility conditions (3.2), the b u,M are subject to the constraint b u,M b v,N η MN = δ uv . (3.26)Furthermore, the u index labels the triplet of SU(2) R and we wish to identify any objects relatedby R -symmetry. This leaves 3 n degrees of freedom in b u,M which is also the dimension of thecoset space ON ,n O(3) × O( n ) . Indeed, we can use b u,M to define a symmetric group element of O(3 , n )by b u,M b uN = 12 ( η MN − H MN ) . (3.27)Because of (3.26), H MN satisfies H MP H NQ η P Q = η MN , (3.28)showing that it is an element of O(3 , n ). H MN can be identified as the generalised metric ofseven-dimensional gauged SUGRA and we will see here that it also becomes the generalisedmetric of the heterotic DFT. Additionally, the scalars | ¯ e | will become the determinant of theexternal seven-dimensional vielbein and d the generalised dilaton of the heterotic DFT.The truncation Ans¨atze for the remaining fields are hA µab i ( x, Y ) = A µM ( x ) ω M ab ( Y ) ρ ( Y ) , hB µν,a i ( x, Y ) = − B µν ( x ) n a ( Y ) ρ ( Y ) , hC µνγa i ( x, Y ) = C µνγ ( x ) n a ( Y ) ρ ( Y ) , hD µνγσ,ab i ( x, Y ) = D µνγσ M ( x ) ω M ab ( Y ) ρ ( Y ) , h e µ ¯ µ i ( x, Y ) = ¯ e µ ¯ µ ( x ) e − d ( x ) / ρ ( Y ) . (3.29)9 .2.2 Consistency conditions and embedding tensor In order to have a consistent truncation one needs to impose a set of differential constraints onthe section ρ , n a , n a and ω M,ab which define the truncation. The so-called “doublet” conditions n a L ˜ ω M ˜ ω N,ab = 0 , L ˜ ω M n a = n a n b L ˜ ω M n b ,∂ ab (cid:0) n b ρ (cid:1) = ρ n a n b ∂ bc n c , (3.30)ensure that the ( , ) and ( , ) representation of the intrinsic torsion vanish. This is requiredin order to avoid couplings to SU(2) S doublets in the SL(5) fields which we want to remove inthe truncation Ansatz in order to have a N = 2 theory.In addition, we require the sections ω M,ab to form a closed set under the generalised Liederivative, i.e. L ˜ ω M ω N ab = 14 (cid:0) L ˜ ω M ω N cd (cid:1) ω P cd ω P ab . (3.31)Given the (3.30) and (3.31) one can identify the object g MNP ≡ L ˜ ω M ω N,ab ω P ab , (3.32)with the embedding tensor of the half-maximal gauged SUGRA. In particular, it has only threeirreducible representations, given by two O( n + 3) vectors f M = n a L ˜ ω M n a , ξ M = ρ − L ˜ ω M ρ , (3.33)and a totally antisymmetric 3-index tensor f MNP = g [ MNP ] , (3.34)which are the only representations allowed by the linear constraint of seven-dimensional half-maximal gauged SUGRA [43]. One can also identify the p = 3 deformation [44] withΘ = ρn a ∂ ab n b . (3.35)It is easy to see that just as in the maximal case, the closure of the algebra of generaliseddiffeomorphisms implies the quadratic constraint of the gauged SUGRA. Thus, imposing thesection condition on ρ , n a , n a and ω M,ab is sufficient to satisfy the quadratic constraints ofgauged SUGRA. Additionally, if one wants to obtain an action principle, one must ensure that ξ M , the so-called trombone tensor, vanishes. Using the truncation Ansatz (3.24), (3.29) onefinds that the dependence on the internal coordinates, Y ab , appears only through the embeddingtensor and as an overall factor of ρ . Thus, we obtain a consistent truncation when the sectionsobey (3.30), (3.31) and the embedding tensor given (3.34), (3.33) and (3.35) is constant and10beys the quadratic constraint. We will now show that the above set-up can be used to reduce the SL(5) EFT to the O(3 , n )heterotic DFT. This requires half of the supersymmetry to be broken which can be achieved byreducing the theory on a generalised SU(2)-structure manifold living in the extended space. Inparticular, we will use the Ans¨atze (3.24) and (3.29) but we will allow the coefficients, which inthe truncation Ansatz only depend on the seven x µ coordinates, to also depend on the extendedspace Y ab . These coefficients will then become the heterotic DFT fields.However, for consistency we will have to impose certain restrictions of their the dependence onthe Y ab . We will see that these conditions allow us to define the n + 3 internal derivatives of theheterotic DFT by “twisting” the 10 extended coordinate derivatives ∂ ab and that the generalisedLie derivative of the SL(5) EFT will reduce to the heterotic generalised Lie derivative. Thegauge group of the heterotic DFT will be determined by the embedding tensor defined by theSU(2)-structure reduction.What we are doing here is reminiscent of the procedure to relate EFT to massive IIA theory[17]. There a reduction was performed on a twisted torus in the extended space and the resultingreduced EFT fields had a limited dependence on the Y ab which upon solving the section conditionled to massive IIA theory (or its IIB dual). We will now use the same Ansatz for the scalars, gauge fields and spinors as in the consistent N = 2 truncation Ansatz of [14], i.e. equations (3.24), (3.29), but still allow the coefficients todepend on some of the extended space coordinates Y ab . Thus we now write the scalar fields as h κ i ( x, Y ) = | ¯ e ( x, Y ) | − / e − d ( x,Y ) / ρ ( Y ) , h A a i ( x, Y ) = 1 √ e − d ( x,Y ) / ρ ( Y ) n a ( Y ) , h A a i ( x, Y ) = 1 √ e d ( x,Y ) / ρ ( Y ) n a ( Y ) , h B u,ab i ( x, Y ) = e − d ( x,Y ) / b u,M ( x, Y ) ω M ab ( Y ) ρ ( Y ) . (4.1)11nd for the gauge fields and external vielbein hA µab i ( x, Y ) = A µM ( x, Y ) ω M ab ( Y ) ρ ( Y ) , hB µν,a i ( x, Y ) = − B µν ( x, Y ) n a ( Y ) ρ ( Y ) , hC µνγa i ( x, Y ) = C µνγ ( x, Y ) n a ( Y ) ρ ( Y ) , hD µνγσ ab i ( x, Y ) = D µνγσ M ( x, Y ) ω M ab ( Y ) ρ ( Y ) , h e µ ¯ µ i ( x, Y ) = ¯ e ¯ µµ ( x, Y ) e − d ( x,Y ) / ρ ( Y ) , (4.2)where will show that ¯ e ¯ µµ is the string-frame vielbein of the heterotic DFT, and the factor of − ρ , n a , n a and ω M,ab to satisfy thesame consistency condition as those required for consistent truncations, see section 3.2. Thereforewe impose equations (3.31) and (3.30) and require that the embedding tensor, given in equations(3.34) and (3.33), as well as the singlet deformations (3.35) are constant and satisfy the quadraticconstraint. To allow for a close comparison to the heterotic DFT formulation in [10, 11, 16] wewill take f M = ξ M = Θ = 0. It is however easy to include these, although when ξ M = 0 we willnot obtain a consistent action principle.However, let us emphasise that we are not performing a truncation because the fields ¯ e , d , b uM , A µM , B µν , C µνρ are still allowed to depend on Y ab . Instead, we are performing a reductionof the theory, which as we will see produces the heterotic SO(3 , n ) DFT with a 7 + 3 split. Thisis similar to the procedure used in [17] to obtain massive IIA from EFT.Recall that the compatibility condition (3.2) implies that b u,M b vM = δ uv , (4.3)and this allows us to define the generalised metric as H MN = 2 b u,M b uN − η MN . (4.4)We see that the b u,M appear exactly like the frame fields in DFT [16, 45]. We will see in section5.1 that indeed the frame formulation appears naturally from the SU(2)-reduction of EFT. Wewill also make use of the left- and right-moving projectors P − MN = b u,M b uN = 12 ( η MN − H MN ) ,P + MN = η MN − b u,M b uN = 12 ( η MN + H MN ) . (4.5) In addition to the differential conditions imposed on the sections ρ , n a , n a and ω M,ab we mustalso impose certain differential constraints on the fields of the reduced theory because these now12arry dependence on the extended space.As discussed in 3.2 and in more length in [14], we must project out the doublets of SU(2) S in order to have a N = 2 theory. Here this now means that we require that the fields of thereduced EFT do not depend on the doublet coordinates, i.e. n a ∂ ab d = n a ∂ ab a = n a ∂ ab b u,M = . . . = 0 . (4.6)This removes the dependence on four coordinates so that the reduced theory is left with a six-dimensional extended coordinate space.Furthermore, we require that the remaining dependence on these ( , ) ⊕ ( , ) coordinatescan be expanded in terms of the sections ω M ab , i.e. ∂ ab = 14 ω abM ω M cd ∂ cd , (4.7)when acting on any of the fields in the reduced theory, e.g. for a vector V M ∂ ab V M = 14 ω abM ω M cd ∂ cd V M . (4.8) O(3 , n ) section condition Because the derivatives acting on the reduced fields can be expanded in the ω M,ab ’s we canintroduce the “twisted” derivatives D M V N = 12 ρ ω M cd ∂ cd V N , (4.9)and as we will see these will become the n + 3 derivatives of the extended heterotic space asin [16]. In order for this identification to work, we require their commutator to vanish, i.e.[ D M , D N ] V P = 0 , (4.10)for a generic object V P of the reduced theory. We can write[ D M , D N ] = 12 ρ L ˜ ω M ˜ ω N P D P −
32 ˜ ω N [ ab ∂ ab ˜ ω M cd ] ∂ cd , (4.11)and using the result of section 3.2.2 we find[ D M , D N ] = f MN P D P −
32 ˜ ω N [ ab ∂ ab ˜ ω M cd ] ∂ cd . (4.12)The first term is thus proportional to the embedding tensor of the extended space while thesecond term is proportional to the section condition acting on the background and the objects13f the reduced theory. We now impose both conditions separately, i.e.˜ ω N ab ∂ [ ab ˜ ω | M | cd ∂ cd ] V N = 0 , (4.13)as well as f MN P D P V Q = 0 , (4.14)where as before V M denotes a generic object of the reduced theory. Given these conditions thederivatives D M commute and thus we can treat them as if they were ( n + 3) partial derivatives.The condition f MN P ∂ P V N = 0 , (4.15)is precisely the condition imposed in heterotic DFT, where f MN P is the embedding tensorencoding the gauge group of the heterotic supergravity [16]. Thus we see the first evidence thatthe embedding tensor of the SU(2)-structure reduction defines the gauge group of the heteroticDFT.We should mention that in the case of massive IIA [17] a similar analysis is used to showthat the reduced theory has a more restricted “section condition”. This implies that the reducedtheory can only contain 10-dimensional solutions, not 11-dimensional ones, as we expect for atheory with a Roman’s mass parameter.Returning to the SU(2)-reduction, it may at first seem strange that we can treat the ( n + 3)derivatives as if they were coordinate derivatives ∂ M even though we started off with only a 10-dimensional extended space. How can all the ( n + 3) derivatives be independent? The answeris of course that they are not but when acting on fields in the reduced theory they can betreated as such because the fields do not have arbitrary coordinate dependence. Their coordinatedependence is restricted by the EFT section condition which we have not yet imposed. This nowtakes the form ∂ [ ab V P ∂ cd ] W Q = 13! ρ − ǫ abcde η MN n e ∂ M V P ∂ N W Q = 0 , (4.16)and similarly for double derivatives, where V P and W Q represent arbitrary O(3 , n ) fields. Thuswe obtain the O(3 , n ) section condition η MN ∂ M f ∂ N g = η MN ∂ M ∂ N f = 0 , (4.17)for any fields f and g of the reduced theory. This implies that the reduced fields can only dependon three coordinates and thus although we formally use the objects ∂ M , only three of these areever non-zero. Let us now show that with the Ansatz described in section 4.1 above, the EFT generalised Liederivative reduces to the heterotic DFT generalised Lie derivative.14e wish to calculate the generalised Lie derivative of two generalised vectors h Λ ab i = Λ M ( x, Y ) ρ ( Y ) ω M ab ( Y ) , h V ab i = V M ( x, Y ) ρ ( Y ) ω M ab ( Y ) , (4.18)where as discussed we now let Λ M ( x, Y ) and V M ( x, Y ) only depend on the external sevencoordinates x µ and the ( , ) ⊕ ( , ) extended coordinates. We then find hL Λ V ab i = 12 Λ M ˜ ω M cd ˜ ω N ab ∂ cd V N + 12 V N ˜ ω N ab ˜ ω M cd ∂ cd Λ M − V N ˜ ω N c [ b ˜ ω M a ] d ∂ cd Λ M + Λ M V N L ˜ ω M ˜ ω N ab = ¯ L Λ V ab + f MN P V M Λ N ˜ ω P ab , (4.19)where we have used (3.34) to write the final term as the embedding tensor defined by the SU(2)-structure, and we have defined¯ L Λ V ab = ˜ ω M ab (cid:18)
12 Λ N ˜ ω N cd ∂ cd V M + 12 V M ˜ ω N cd ∂ cd Λ N (cid:19) − V N ˜ ω N c [ b ˜ ω M a ] d ∂ cd Λ M . (4.20)The first two terms are just12 Λ N ˜ ω N cd ∂ cd V M + 12 V M ˜ ω N cd ∂ cd Λ N = Λ N ∂ N V M + V M ∂ N Λ N , (4.21)where as we discussed above we defined ∂ M = 12 ρ ω M ab ∂ ab . (4.22)For the last term we use (3.22) to write ω M ab ω N ca ω P bd = − ω N cd δ MP + η NP ω M,bc . (4.23)Hence we find¯ L Λ V ab = ˜ ω M ab (cid:16) Λ N ∂ N V M + V M ∂ N Λ N − V ( M ∂ N Λ N ) + η MP η NQ V N ∂ P Λ Q (cid:17) = ˜ ω M ab ¯ L Λ V M , (4.24)where ¯ L Λ V M = Λ N ∂ N V M − V N ∂ N Λ M + η MP η NQ V N ∂ P Λ Q , (4.25)is precisely the SO(3 , n ) generalised Lie derivative with no gauging. Putting everything togetherwe obtain hL Λ V ab i = ˜ ω M ab L Λ V M , (4.26)15here we defined L Λ V M = Λ N ∂ N V M − V N ∂ N V M + η MP η NQ V N ∂ P Λ Q + f NP M V N Λ P . (4.27)This is the generalised Lie derivative of the heterotic DFT, with gauge group defined by theembedding tensor f NP M , see equation (3.6) of [16].
We will now show how the EFT action reduces to that of the heterotic DFT [16] with a Kaluza-Klein split with seven external dimensions [22].
Using the Ansatz (3.24) it is easy to show that the EFT scalar potential reduces to the scalarpotential of the heterotic DFT. Let us begin by evaluating the intrinsic torsion with the Ansatz(3.24) and using (3.34), (3.33) and (3.35). As we discussed earlier we take f M = ξ M = Θ = 0to allow for a close comparison with [16]. We find that the doublets of SU(2) S vanish exactly asrequired. The remaining irreducible representations become h S i = 12 ρ e − d/ Θ , h T i = 16 ρ e d/ ǫ uvw Ω uvw , h T u i = 1 ρ √ e d/ (cid:20) Ω u + 47 b uM ∂ M ln | e | (cid:21) , h S u i = √ ρ e d/ (cid:20) Ω u + 67 b uM ∂ M ln | e | (cid:21) , h T ab i = 128 ρ √ P M + N ω N ab ∂ M ln | e | , h T uab i = 112 ρ √ ǫ uvw e ¯ uM ω M,ab Ω vw ¯ u . (5.1)Here we have defined the generalised coefficients of anholonomy Ω uvw as in [16] (see also [11],[18], [45]). That is, Ω uvw = (cid:0) L b [ u b vN (cid:1) b w ] N , (5.2)in terms of the generalised Lie derivative of the heterotic DFT including the gaugings, i.e. (4.27),and Ω u = ∂ M b uM − b uM ∂ M d . (5.3)Furthermore, e ¯ uM , with ¯ u = 1 , . . . , n represent the n right-moving vielbeine of the generalised16etric satisfying P M + N e ¯ uN = e ¯ uM , P MN + = e ¯ uM e ¯ vN η ¯ u ¯ v , (5.4)with η ¯ u ¯ v an O( n ) metric which is not necessarily constant, and Ω uv ¯ u is similarly defined in termsof these generalised vielbeins, see [16].Plugging (5.1) into (3.12) and using the Ansatz for the metric (3.29) in (3.14) we find theheterotic DFT scalar potential in the frame formulation [16]. h| e | V i = ρ | ¯ e | e − d (cid:20)
112 Ω uvw Ω uvw + 14 η ¯ u ¯ v Ω uv ¯ u Ω uv ¯ v + 12 Ω u Ω u + Ω u b uM ∂ M ln | e |− H MN ∂ M g µν ∂ N g µν (cid:21) . (5.5) We have already shown in section 4.4 that the EFT generalised Lie derivative reduces to that ofthe heterotic DFT. This means that the external covariant derivative of the EFT [3], defined as D µ = ∂ µ − L A µ , (5.6)will also reduce to the heterotic external covariant derivative, see e.g. [22] for the O( d, d ) versionthereof.For example, we find h D µ B u,ab i = ω M,ab h ∂ µ (cid:16) b uM e − d/ (cid:17) − L A µ (cid:16) b uM e − d/ (cid:17)i , h D µ A a i = 1 √ n a (cid:16) ∂ µ e − d/ − L A µ e − d/ (cid:17) , h D µ κ i = ρ h ∂ µ (cid:16) | ¯ e | / e − d/ (cid:17) − L Aµ (cid:16) | ¯ e | / e − d/ (cid:17)i , (5.7)where one can read off L A µ (cid:16) e − d/ b uM (cid:17) = A µN ∂ N (cid:16) e − d/ b uM (cid:17) − e − d/ b uN ∂ N A µM + e − d/ b uN ∂ M A µN + 15 e − d/ b uM ∂ N A µN + f NP M A µN b uP ,L A µ e − d/ = A µN ∂ N e − d/ + 25 e − d/ ∂ N A µN ,L A µ (cid:16) | ¯ e | / e − d/ (cid:17) = A µN ∂ N (cid:16) | ¯ e | / e − d/ (cid:17) + 15 | ¯ e | / e − d/ ∂ N A µN . (5.8)17hese equations imply that L A µ b uM = A µN ∂ N b uM − b uN ∂ N A µM + b uN ∂ M A µN + f NP M A µN b uP ,L A µ e − d = A µN ∂ N e − d + e − d ∂ N A µN ,L A µ | ¯ e | = A µN ∂ N | ¯ e | . (5.9)Thus, we can see that b uM transforms as a O(3 , n ) vector field, e − d as a scalar density of weight1 and | ¯ e | as a scalar, exactly as in the heterotic DFT.The same computation for the full external vielbein and the gauge fields shows that ¯ e µ ¯ µ transforms as a scalar, A µM as a O(3 , n ) vector and B µν and C µνρ as scalars with respect to theheterotic generalised Lie derivative. Thus we see that the EFT covariant derivative D µ reducesto that of the heterotic DFT which we label by D µ = ∂ µ − L A µ . (5.10)We can now compute h D µ A a i = − √ e − d/ n a D µ d , h ( D µ B u,ab D ν B ucd ) ǫ abcde A e i = 2 √ D µ b uM D ν b uM + 24 √ D µ d D ν d . (5.11)Using ∂ µ H MN ∂ ν H MN = 8 ∂ µ b uN ∂ ν b uN , (5.12)we can rewrite the second equation as h ( D µ B u,ab D ν B ucd ) ǫ abcde A e i = √ D µ H MN D ν H MN + 24 √ D µ d D ν d . (5.13)Thus, the scalar kinetic terms reduce to h| e | L SK i = h| e | g µν (cid:18) √ D µ B u,ab D ν B ucd ǫ abcde A e − D µ A a D ν A a (cid:19) i = ρ | ¯ e | e − d (cid:18)
18 ¯ g µν D µ H MN D ν H MN + 4¯ g µν D µ d D ν d (cid:19) , (5.14)which are the the scalar kinetic terms of the heterotic DFT in a Kaluza-Klein split, see e.g. [22].Let us now consider the reduction of the field strengths using (3.29). We find hF µν ab i = ρ ω M ab F µν M , hH µνγ a i = − ρ n a H µνγ , hJ µνγσa i = ρ n a J µνγσ , (5.15)18here F µν M , H µνγ and J µνγσ are the reduced field strength of the Kaluza-Klein split heteroticDFT [22], i.e. F µν M = 2 ∂ [ µ A ν ] M − [ A µ , A ν ] MC − ∂ M B µν ,H µνρ = 3 D [ µ B νρ ] + 3 ∂ [ µ A νM A ρ ] M − A [ µM (cid:2) A ν , A ρ ] (cid:3) C,M ,J µνρσ = 4 D [ µ C νρσ ] + ∂ M D µνρσ,M . (5.16)Here [ A µ , A ν ] MC = 12 (cid:0) L A µ L A ν − L A ν A µ (cid:1) , (5.17)is the antisymmetrised heterotic generalised Lie derivative. Because the three form decouplesfrom the two-form, it is not necessary to include it in the Kaluza-Klein split DFT tensor hierarchy.With the above reduction it is easy to see that h| e | L kin,vectors i = ρ | ¯ e | e − d ¯ g µγ ¯ g νσ F µν M F γσN (2 b u,M b uN − η MN )= − ρ | ¯ e | e − d ¯ g µγ ¯ g νσ F µν M F γσN H MN , (5.18)which is the correct kinetic term for the vector fields. Similarly, we reduce the kinetic term forthe two-form potentials L kin, = − H µνρ,a H µνρb A a A b , (5.19)to find h| e | L kin,2-form i = − ρ | ¯ e | e − d ¯ g µσ ¯ g νρ ¯ g γλ H µνγ H σρλ , (5.20)again reproducing the correct kinetic term for the two-form potentials.Finally, it is easy to see from [14] that the topological term vanishes in the reduction. Thusneither the three-form potential nor its four-form field strength appear in the action as required. Let us now use the results presented here and in [14] to discuss the M-theory / heterotic dualityin the context of the SL(5) EFT. Consider the SL(5) EFT with a K3 surface in the extendedspace. We can now perform a consistent truncation in two ways.We let i, j = 1 , . . . , Y ab as Y i . We further let ρ ω M,ij be the 22 harmonic two-forms of the K3surface. Furthermore we take n = n = 1 and ρ = constant. Thus we have that˜ ω M ab = ρǫ abcde ω M,cd n e , (6.1)19ave as their only non-zero components˜ ω M ij = 1 √ ρǫ ijkl ω M,kl . (6.2)From the generalised Lie derivative (2.2) we thus find L ˜ ω M ω N,ij = 0 , (6.3)because ∂ ij ˜ ω M,kl = 0 and L ˜ ω M ω N,i = 1 √ ǫ klpq ω N,ik ∂ l ( ω M,pq ρ ) = 0 , (6.4)because ρ ω M,ij are harmonic. We also find that L ˜ ω M n a = 0 , ∂ ab n b = 0 , (6.5)and so the doublet and closure condition are satisfied and the embedding tensor and the singletdeformation vanish.We can choose the four coordinates Y i as parameterising our section in which case we seethat we have performed a consistent truncation of 11-dimensional SUGRA on K3. This way wehave obtained an ungauged seven-dimensional SUGRA with 19 abelian vector multiplets, exactlyas required.However, there is also another interpretation of the above set-up. We could have first per-formed a reduction of the SL(5) EFT on K3 and obtained the SO(3 ,
19) heterotic DFT withabelian gauge group. The DFT fields would then be required to depend only on the six coordi-nates Y ij since n b ∂ ab = ( ∂ i , . (6.6)is required to vanish for all reduced fields. Furthermore, we would have obtained 22 twistedderivatives ∂ M = 12 ˜ ω M ab ∂ ab = 12 ˜ ω M ij ∂ ij . (6.7)Now we could have performed a trivial toroidal truncation of this heterotic DFT, in order tomatch the previous set-up, where we would have chosen three of the Y ij as parameterising oursection. This would have described the consistent truncation of the heterotic string on T withthe gauge group broken to its abelian subgroup.From this perspective the difference between the two cases, M-theory on K3 and heteroticon T resides in the choice of section: if we take the four coordinates of the K3 surface as oursection then we are performing a consistent truncation of 11-dimensional SUGRA on K3, while ifwe choose one of the other six coordinates as our section then we have more naturally performeda T truncation of the abelian O(3 ,
19) heterotic SUGRA.20
Conclusions
In this paper we have shown that exceptional field theory not only contains 11-dimensionaland IIB SUGRA but also the heterotic SUGRA via its doubled version, the heterotic DFT.The EFT can be reduced to the heterotic DFT when its extended space contains a generalisedSU(2)-structure manifold. The reduction used is very similar to a truncation on generalisedSU(2)-structure manifolds [14]. However, the coefficients appearing in the expansion of the EFTfields are still allowed to depend on the extended space, albeit subject to further constraintswhich ensure that the resulting theory has N = 2 SUSY. The embedding tensor defined by thisreduction procedure defines the gauge group of the heterotic DFT while the number of sectionsof the ( , )-bundle of SU(2) S × SU(2) R ⊂ SL(5) used in the reduction defines the number ofvector fields of the heterotic DFT.We have shown that using the reduction Ansatz it is natural to introduce n + 3 “twisted”derivatives which commute when the reduced fields are subject to a number of constraints andthus can be thought of as n + 3 coordinate derivatives. The constraints imposed are also re-quired from the heterotic DFT perspective directly [16]. Furthermore, with this Ansatz thegeneralised Lie derivative of the EFT reduces to that of the heterotic DFT. This suggests thatone should interpret the n + 3 “twisted” derivatives as the Fourier duals to momentum andwrapping modes. Indeed, in the case of the consistent truncation of M-theory on K3, we find 22wrapping derivatives this way.Finally, we have shown how the duality between M-theory on K3 and the heterotic string on T arises in EFT. In the M-theory case, the K3 surface is taken to form the four-dimensionalM-theory section of EFT, whereas in the heterotic picture, it is a subset of three out of the extrasix directions which are taken as the section. Thus the duality here is generated by a changeof section even though there are no isometries. It is thus an example of a “generalised duality”without isometries.It would be interesting to further explore the heterotic / M-theory duality, in particularwhether it can capture gauge enhancement when two-cycles of the K3 surface shrink. For exam-ple, in [46] it was shown that double field theory can capture the gauge enhancement at self-dualcircles. This would be an interesting test to see whether EFT really captures phenomena whichgo beyond SUGRA but are related to wrapping sectors of M-theory. Another thing to under-stand would be whether α ’ corrections are correctly handled in this picture, see for example [47]for ways these corrections can be incorporated in generalised geometry.Furthermore, one might ask what happens in lower dimensions. For example, one wouldexpect to see the duality between IIA and the heterotic string in the Spin(5 ,
5) EFT [48, 49]. Infour dimensions one would hope to see mirror symmetry arise between consistent truncations onexceptional SU(6)-structures [50].Yet another possible avenue of further research would be to study solutions of exceptionalfield theory on K3. In [19], [20] it was shown that the string and pp-wave solutions are unified asone solution of DFT and similarly for branes and monopoles in EFT. How does one describe a21olution of EFT on K3 corresponding to an M5-brane wrapping the K3? In this case, its heteroticDFT dual should be the heterotic string and it would be interesting to see how this arises fromthe formalism presented in this paper.
The author thanks David Berman, Diego Marqu´es, Carmen Nu˜nez, Felix Rudolph and HenningSamtleben for helpful discussions. The author would also like to thank IAFE Buenos Aires forhospitality while part of this work was completed. This work is supported by the ERC AdvancedGrant “Strings and Gravity” (Grant No. 320045).
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