Jhep01(2016)132

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JHEP01(2016)132 Springer January 8, 2016 spaces, Super- January 21, 2016 : December 15, 2015 exceptional ﬁeld : : is independent, and 7(+7) E 10.1007/JHEP01(2016)132 Accepted 7(+7) Published Received = 4 superspace enlarged by counterpart of the) general E doi: f D central charge ests that only a part of the section ge of the maximal supersymmetry = 8 e Country UPV/EHU, N exceptional ﬁeld Published for SISSA by = 8 N 7(+7) E . 3 1512.02287 The Authors. c Supersymmetry and Duality, Extended Supersymmetry, Super

We study the properties of section conditions of the , [email protected] E-mail: Department of Theoretical Physics, UniversityP.O. of Box the 644, Basqu 48080 Bilbao,IKERBASQUE, Spain Basque Foundation for48011, Science, Bilbao, Spain gravity Models ArXiv ePrint: Keywords: we show that thissolution part of indeed the suffices strong to section obtain condition. the (classical superalgebra. In particular, the superparticleconditions model corresponding sugg to generators of SU(8) subgroup o theory from the perspective ofadditional superparticle bosonic model coordinates in related to the central char On section conditions of theory and superparticle in Open Access Article funded by SCOAP Abstract: Igor Bandos superspace JHEP01(2016)132 ] 4 5 7 8 1 4 7 9 12 25 – 14 tions of . In both Σ y coordinates, which D licated, but also more e manifest the U-duality f the origin of these mys- ace t T-duality is provided by f section conditions). An tions that all the physical ns are imposed on any pair ]. nic coordinates ]), is formulated in the space is restricted by the so-called ponding section conditions is ries by introducing additional acterized, at least partially, by 27 12 a very special gauge symmetry , g. [ ] designed to have a manifest T- 26 11 – 1 exceptional field theories” (EFTs) [ ) n ( of the complete set of 2 n – 1 – E D exceptional field theory = 8 central charge superspace duality symmetry the number of spacetime coordinates ) n N ( 7(+7) n E E ] one develops “ 13 the strong version of which is imposed on a pair of any two func In the case of double field theory the strong section conditio A progress in comprehension of structure and clarification o In ‘exceptional field theory’ the situation is a bit more comp 2.1 Section conditions2.2 of Central charge superspace 3.1 De model Azcárraga-Lukierski 3.2 Improved superparticle model in central charge supersp (‘gauge coordinate transformations’) was developed in [ of two functions offields 2D depend coordinates only and on are Dthe solved of freedom by 2D in bosonic the choosing coordinates. condi is the The often set manifes referred of to these interesting approach as relating choice section of conditions the of DFT section to (hence the name o terious conditions are one of the main goals of this paper. the theory. cases the dependence ofsection the conditions fields on additional coordinates illustrative. To make manifest duality symmetry, characteristic for stringwith theory (see doubled e. number ofsymmetry (D=10) of spacetime M-theory coordinates. [ which To are mak also formulated in spacetime with additional boso is generically more thenmore doubled obscure. and the origin of the corres bosonic coordinates. Double field theory (DFT) [ 1 Introduction The present stage of development ofexploiting String/M-theory the is char possibility to make manifest duality symmet 4 Superparticle and section conditions 5 Conclusion and outlook 3 A superparticle model in Contents 1 Introduction 2 Preliminaries JHEP01(2016)132 ) by 2.2 = 8 (1.1) (1.4) (1.3) (1.5) , was ] was ] (see . itional ), ( N D 21 32) 7(+7) 28 | , E 2.1 (60 oordinates, ] to replace 19 . of , central charge 28 ij Π uper)fields are s equivalent to ¯ Z 17 Λ ˙ β E ) of the maximal ˙ ates of standards α t = 4 + 56 bosonic ij ǫ = ,Z ), which can be called aning of section condi- ij } thors of [ ij ¯ Z ˙ ¯ EFT of [ βj y , ¯ Q e section 2, eqs. ( ij , = ( ion in the standard icle model in central charge tant and illustrative case of e. In it one imposes, instead y ˙ αi ravity multiplets and their 28 tions of the EFT. In addition, Σ = 8 supergravity in [ SU(8) covariant derivative 7(+7) ¯ Q Z = ( E × { ) N Σ C y , , = 4 , . . a P D ˙ β , and the direct product symbol reflects the = 0 = 0 a α (related to the generators ) = 0 56 matrix Ξ (symplectic ‘metric’) is antisym- σ Π Π ΠΣ , i.e. the conditions ) = 0 (1.2) i j ∂ ∂ ... δ Σ E × ΣΠ – 2 – ( t y ... ⊗ ⊗ E ) = 0 with SL(2 Π ( = t = ∂ Σ Σ Σ } ... Σ ∂ ( ∂ ∂ ˙ ∂ βj Σ ΣΠ ¯ D ΣΠ ΣΠ Q E ΣΠ , t E Ξ corresponding to this superalgebra is called t E i α t Q ]. 32) { | 32 0 (60 . The curved version of central charge superspace, Σ , strong section conditions ij while 133 matrices Z ) and the first observation beyond our approach is that the add can be associated with central charges ] to be independent on ] uses the condition αβ ]) by the bosonic supervielbein forms corresponding to new c Σ ΠΣ ) are symmetric, ǫ Σ 29 29 Ξ Π 31 , y y , = With this condition the model in central charge superspace i Λ , µ − x E } ). To remove the unwanted extra degrees of freedom, all the (s 1 28 t 30 = j β ij ¯ . The set of their coordinates contains, besides the coordin y = 8 supersymmetry algebra = 8 supergravity [ ΣΛ = 4 superspace, also 56 bosonic coordinates , ,Q ΣΠ ij i α ]). d effective field theory which is formulated in the space with 60 N N y ), the following Q = Ξ 29 { From the above perspective one can state that the The use of (curved) central charge superspace allowed the au It this paper we try to understand better the structure and me 1.2 To be precise, [ = ( = 8 1 = 4 = 4 ΣΠ 7(+7) Σ E D central charge coordinates used for alternative superspace formulation of The supergroup manifold Σ N superspace superspace [ We refer on the main text for the notation and more details (se fact that these conditions arethe applied weak to section any conditions pair of the func t metric Ξ assumed in [ 56 coordinates and below) and just notice here that the 56 formulated in a bosonicof body ( of the central charge superspac D the superform generalization of 28magnetic gauge duals fields (which of both the superg appear in supergravityy formulat also [ tions by approaching themsuperspace. from the We perspective restrict ofE ourselves superpart by a particular but impor coordinates ( are imposed. JHEP01(2016)132 ) 0. for 1.6 ≡ (1.6) (1.9) (1.7) (1.8) n ’s are ) and (i.e. a r Π e weak p ≤ p ]. . Then, 1.7 Σ ˜ n n 40 p ΣΠ Below we will section as a condition ,..., , schematically r 2 y = 1 r ˜ n = 7 (we will discuss , metric, Ξ r sidered, the momenta n p hanic relation ( ctation is that ˜ rong section condition is till now. ,..., = rpart of the strong condi- ) of the section condition. serves as prototypes of the es) n stions obtained by studying ) (and assuming that = 1 on more than 11 coordinates. Σ 1.5 cal’ counterpart of the condi- eful. y ) we arrive at the solution ( 1.8 M-theoretic grounds one should nd related issues. Σ by ˜ n ) by replacing the momentum by y n beyond the recent study in [ 1.7 , r ,..., r ˜ y ∂ ∂ . = 1 of ‘internal’ coordinates ˜ , n = = 0 Σ r ) is satisfied identically due to antisymmetry Π i∂ – 3 – p 2.2 , r ] the solutions with ˜ Σ r p 21 7→ − p , ∂ r , Σ ) Σ ΣΠ p 0. 19 E ) with a prescription of ( K Moreover, on this way, one can help to find the general ... t , ( ≡ r 3 = 1.9 17 ∂ Π . A possible choice of this latter defines a r Σ ∂ r p Σ Σ of central charge coordinates Σ ∂ K r K y EFT [ ΣΠ = 6 are found and shown to correspond to the embedding of D=11 ) = are commutative canonical variables, both the strong and th n 7(7) Σ ... ( E p matrix Σ ∂ ˜ n . As × ) conjugate to the central charge coordinates ) as a (first class) constraint and impose its quantum version Σ ij ∂ 1.7 , p ) is satisfied identically due to antisymmetry of symplectic ) while the weak counterpart of ( ij p 2.2 1.3 In the case of However, let us imagine that we first solved the classical mec When particle mechanics in central charge superspace is con The general expectation is that the result of imposing the st If we perform a straightforward ‘quantization’ of ( The author thanks Henning Samtleben for a discussion on this a Probably the understanding of this fact was also a motivatio = (¯ 2 3 Σ section conditions are represented in superparticle model derivatives Again, as in thetions case ( of weak section conditions, the ‘classi any solution of the strongnot section expect conditions: to have on a physical/ However, solution to allowing physical our fields best depend knowledge, this fact has not been proved with some 56 conjugate to some subset of the strong section conditions. shown that its general solution have the form derivative, on the wave function, we clearly arrive at the weak version ( superparticle in central charge superspace happens to bep us show that this is indeed the case. In achieving this the sugge of symplectic ‘metric’, Ξ consider ( and D=10 type IIB supergravity into the EFT. The general expe tion ( that all the fields depend on a smaller number ˜ are obeyed by any function of EFT. Notice that this is a counte with some set of independent (or constrained among themselv particular solution of the section conditions). this below) and with ˜ performing the quantization of ( JHEP01(2016)132 , ] f – ). 32) 33 28 21 | , 1.4 Σ = (2.1) (2.2) (2.3) o the 0 (60 , that , 19 Π , 32) , can be , | EFT and Σ symmetry 17 y (60 ) and ( ) modulo the 1.3 7(+7) representation, 7(+7) 1.9 E supergravity [ E part of the section ). Moreover, it is of extended space- 56 arge coordinates is Σ 1.7 y . Already in early 80th )( ric mechanics models in ow on we will call them charge superspace Σ ETF these read [ = 0 involving the generators 7(+7) ound in curved Σ this for the s a manifest E Π tion of the complete set of the exceptional field theories. p sorial central charge coordinates [ . on of the type ( 7(+7) ) Σ generators in group, suggested us that probably E n ) is given above in ( p ( nfinite set of tensorial coordinates which n , and the remaining 56, µ 2.2 , . ΣΠ E x 7(+7) . ] which we also discuss below. The fact H 7(+7) t ), ( E E 42 = 0 = 0 ), 2.1 = 0 2 2 1.7 F F are . It will be interesting to understand whether ]. F Π Π Π 39 ∂ ∂ – 4 – Σ ∂ , . In the case of Λ 1 1 Σ 38 7(+7) ∂ F F G exceptional field theory t E Σ Σ ΣΠ ∂ ∂ G also obeys a weak section condition = SU(8) of the t ΣΠ ΣΠ 7(+7) G Ξ F H t proposal [ ). It can be considered as an improved version of model where E 11 Σ 1.7 4 E Λ G exceptional field theory are the conditions corresponding t ] is defined in spacetime with 60 = 4+56 bosonic coordinates, o t 21 is the invariant tensor of Sp(56) (symplectic ‘metric’), Λ ΠΛ , 7(+7) 19 ΛΠ E , = Ξ strong section conditions Ξ = SU(8) subgroup of are arbitrary ‘physical’ functions of the coordinates 17 − 2 H ] in the frame of F ΣΠ = G 37 133. A compact way of writing eq. ( t ). We show that this is indeed the case so that the independent and had been introduced in the superfield approach to maximal D=4 which might generate ‘classical section conditions’ (CSCs EFT) [ ΠΛ 1.7 1 Σ 56; ,..., F y 32) In this paper we present a superparticle model in flat central The above discussion motivated us to search for supersymmet In the EFT framework the dependence of the fields on central ch | These coordinates can also be extracted from the set of D=11 ten 4 7(+7) = 1 ] as coordinates of central charge superspace, so that from n ] which in their turn can be considered as finite subset of the i (60 ,..., E such considered as a vector in the fundamental representation of time, Ξ interesting property should be preserved in flat superspace the general solution ofCSCs this ( part gives also the general solu ( G this situation is also reproduced in the case of other 2 Preliminaries 2.1 Section conditions of As we have already said, the exceptional field theory which ha generators of of a maximal compact subgroup Here In EFT every physical function ‘central charge coordinates’. restricted by the which 4 are the usual D=4 spacetime coordinates, conditions of the proposed by de and Azcárraga Lukierski inthat 1986 the [ model produces only a part of CSCs ( SU(8) transformations. not difficult to conclude that if such a hypothetical model is f appeared in [ 1 which generates a part of ( 36 29 solution of the strong section conditions. Below we will find show that this general solution is described by the expressi Σ JHEP01(2016)132 = and (2.4) (2.5) (2.7) (2.8) (2.6) (2.9) ΣΠ (2.10) 28 genera- ; in par- 63 ] it was also ; clearly the . + 7(+7) F 8 19 E 70 , . . by = 17 he l ,..., i = SU(8)-generators 32) | δ = 0 133 k i ), reads H j i = 1 (60 ) as δ ]. In [ hat the natural splitting 2.2 − 19 1.3 l , . , j dditional coordinates. avity in the exceptional field 6 additional bosonic coordi- δ , t he symplectic metric Ξ 17 pace Σ k i i j ons ( = 0 = 0 δ , i, j t ibes embedding of the standard , ′ l kl ij ′ ) = 2 1 ¯ k y ∂ ˇ α † ¯ . ∂ ) = = ¯ ⊗ j , ϑ i l ⊗ ∗ Σ ] t G/H kl ′ ) j ( t = 0 can be decomposed on the sum of j ∂ δ ′ , y ij j i k i ij y i ¯ m ∂ [ δ ( ′ ∂ can be decomposed as δ l x 1 4 ′ ) allow for dependence of all the fields on 7 , central charge coordinates. The corresponding 7(+7) k , ⊗ ′ splits onto the set of 63 = j − – 5 – E ) Σ ′ 2.8 ij j y kl , ) = ( ¯ 7(+7) i ∂ ik pqrs y ˇ α t )–( ∂ E , t ji ijkli ij − 7(+7) ǫ y ¯ , ϑ ∂ ⊗ 2.9 E ij 1 exceptional field theory [ ijkl m − 4! ¯ t ∂ jk X = ¯ = ∂ − ijklpqrs ⊗ ] ǫ SU(8) generators , G ij -part of string section condition, ( + / kl ij 7(+7) = ( ) = ( 4! ) ∂ ∂ / E H jk ij ) with both derivatives applied to one function Sp M ¯ ∂ ⊗ , t ]). Below we will show that there are no solution of the strong 7(+7) , ∂ Z 2.1 = 1 representation of ij ⊗ ) is satisfied identically. ij [ E , y 23 ¯ ∂ ∂ ) G/H ik = 56 t ijkl 2.2 representation of ∂ ij ¯ t ] in the ¯ y = ( , 41 = ( = Σ ij 133 † ∂ y G/H G . Hence the ) ∂y t = ( ijkl 28 := t × 0 Σ ( y Σ 28 I and 70 ∂ is the subalgebra of Sp(56) and the above representation of t j 28 i t × 0 The adjoint The fundamental The strong section conditions ( 28 representations of its maximal compact subgroup SU(8), so t = I − 7(+7) H decomposition of derivatives is tor is given for the case of the following representation of t weak counterpart of ( E 2.2 Central charge superspace We denote the local coordinates of the central charge supers additional coordinates; the11D corresponding supergravity solution [ descr described a solution ofnates, section which corresponds with to functions embeddingtheory of depending (see 10D on type also II recent supergr section [ conditions with functions depending on more than 7 a This splitting was implied when we called of the set of additional bosonic coordinates is Using this splitting we can write the strong section conditi t 28 This is a counterpart of ( ticular the set of generators of JHEP01(2016)132 . ij = 0, ¯ ∂ = 0), i ), the (2.18) (2.16) (2.17) (2.11) (2.12) (2.13) (2.21) (2.14) (2.19) (2.15) (2.20) ab j . , j 1.2 ∗ = 2 ) , ) ˙ α ij ˙ ˙ αi β ˙ ¯ αi ij Z ¯ ¯ θ ¯ E ∂ , d ˙ j e found in (the α . α ¯ i θ = (Ω 8 = and E − e torsion two forms α , ˙ . ij , αi a ¯ i = ( A ∂ 8 E i∂ j ab β ,..., ˙ , α α SU(8), σ ) = 0 were used instead ) Ω BC he conditions ( ase of flat central charge α j a T ab = 2 × ∧ . = 1 σ E ... B ,..., i Ω ( ) ij α θ ˙ ∧ ] E αj 1 4 can be set to zero, Ω ij ( C j Z j , 2 1 θ E = 1 ∧ i D ,E , = i , ,, α [ a C Ω ) − ] − α k j ij E ] i∂ ˙ α β − k j ˙ ¯ idθ , i , i, j ˙ αi 2 1 E , β ¯ α Ω ∂ = E , ) of 56 additional coordinates on two 2 2 i Ω a − β , , ij a b , ∧ ∧ − Ω ∧ ij ) = 2.4 P i Ω k α [ i representation of SU(8) and we can use ¯ y i ,E ) = 0 and ˙ k ∧ ˙ [ αi = = 1 = 1 βi d ∧ a dθ β i E ˙ ˙ b E ˙ ... α α E 28 αi ,T = ( E = E ¯ i α Q − ij ij + 2 + 2Ω ) with α i − T = ( − ¯ ˙ D – 6 – ; E αi and ij ij = 4 supersymmetry generated by supercharges a α i a , , 1.1 ij ¯ E ) is the SU(8) connection superform (Ω ¯ 2 2 T by dE D , dE d dE , , 28 , dE Z ,E ( ij ij = i = = i = = 32) j ∂ | ¯ θ = 1 = 1 = 8 ˙ αi α i a ij ij ,T ,E d αj M a ¯ (60 a ) E E E E E θ N Ω T α σ i D D D D D − θ M ,E a i 2 = = ( a Z := := := := ∂ , α , α ] (in this latter d α E ) + , 3 a α i ) ˙ A ij ij ) is the spin connection, and Ω αi , 29 i ] i ˙ ¯ ˙ αi T T j T α ¯ 2 ¯ = θ T θ E Z T ¯ θ ¯ , θ a a ( , i 1 D ∗ ) = ( [ σ σ α i , ) i ˙ ab M ( α j θ Z = ¯ i θ 2 1 Ω ( ] and [ dθ = 0 A i id 2 A (Ω M M = ( − 28 T a SU(8) covariant derivatives of the supervielbeins define th i − E − ˇ α Z α − d ⊗ ϑ a ij i∂ M ) = = − Z = 4 supergravity as a model in central charge superspace can b C dx dy i , d j ba = D = = )). i α := a ij Q For (most of) our purposes here it is sufficient to discuss the c The torsion constraints describing, when supplemented by t = Ω E = 0, and the fermionic coordinates carry indices of SL(2 A E 1.2 = 8 j E ab i Ω where Ω superspace. In the flat limit the connections are trivial and of ( the following set of supervielbein forms conjugate sets carrying indices of the Ω N appendices of) [ and the supervielbein forms of Σ the superalgebra of which has the form ( These are invariant under rigid The SL(2 In this case it also makes sense the decomposition ( JHEP01(2016)132 . 32) | ]. In (3.2) (3.5) (3.1) (3.4) (2.23) (2.22) (2.24) 0 (60 by the 42 32) | t section 4, (60 : . 32) | ij τ (60 ˆ ¯ E , ij τ ˙ ij τ ˆ β E ˙ ˆ α E ǫ . ˙ thers) the following action q βj b oup manifold associated to E ij τ = cussion below) and ˆ ¯ improved version, which does ∧ E )) (3.3) ill analyze the interdependence τ ij ij τ ˙ n. αi ( ) in the case of flat superspace ˆ E ˆ Z , iE ( q 2 2.19 A − dτ m , p = 8 central charge superspace Σ M = 0 = ) (( E Z ˙ ij N ) αi is a constant obeying ij τ b τ ˆ ¯ = 8 superspace enlarged by central charge b ( E 2.11 4 = 2 ij τ ij τ + 2 M which is described as a line in Σ / ˆ ¯ N ˆ E 2 ˆ E – 7 – Z 1 ,, aτ ,T d ˙ m ˆ q β E W = 8 central charge superspace a α b a τ N = αβ = 4 central charge superspace Σ σ ˆ ǫ E a = τ ˙ βi β = j N ˆ , D E ¯ q ij 2 E E ) ¯ b p dτ ij ∧ ∧ dτ ), i.e. of the flat = 8 α α i i , p = Z ) can be also called Cartan forms as they obey the torsion 1.1 ij N A ,T iE iE , ¯ p ). m ˆ , − − E τ a 2.19 ( aτ = p ˆ E = = := 0 M S aτ a τ ˆ ˆ = ( Z a ˆ α i E ij E , was proposed by de and Azcárraga Lukierski in section 2 of [ ¯ T a T T τ q ˆ E 32) L | ∂ m 0 (60 -symmetry without imposing any condition by hand. In the nex = := κ a a p p is a constant with dimension of mass, ) to the superparticle worldline ] and Joséde Azcárraga Jerzy Lukeirski discussed (among o m 32) It is convenient to introduce the canonical momenta The supervielbeine ( | 42 0 (60 (the reason to impose this relation will be clear from the dis is the pull back of the supervielbein form ( coordinates, Σ using suggestions provided by this superparticleof model the we section w conditions and present their general3.1 solutio De model Azcárraga-Lukierski In [ An interesting superparticle model in flat In it 3 A superparticle model in coordinate functions which can be identifiedthe with supersymmetry structure algebra ( equations of a supergr Σ this section, after reviewingpossess this the model we will present its ‘constraints’ with constant coefficients for massive superparticle in JHEP01(2016)132 ) . ) ext 2 (3.9) (3.7) (3.6) (3.8) 3.13 m (3.10) (3.13) (3.14) (3.12) (3.11) − a ) between p a ˙ αj 3.2 p ( ¯ κ e ij 2 1 p e ot need imposing c.c. . 2 dτ -symmetry. + − . κ α i Z i α κ δθ − , ˙ αα posing the condition ( a )) y the following first order j )) = 0 i ˜ σ τ after making this interesting τ αj τ δ ( e present perspective one can a ( ˆ 2 ˆ p E ˆ Z 4 Z ( ij ( m . = p a α 2 , the supervielbein forms are given , + ˙ αi M M ) m ¯ θ + 2 ). E E kj 32) ˆ κ | ) ) Z p α ( . ) can be written in the form τ τ ) a ik ( ( 0 (60 i τ the momenta in central charge directions 4 ¯ 3.12 p B ˆ ¯ / E 3.1 M M α 4 = 2 2 a ˆ ˆ T i / Z Z , δ σ j α m ǫ ǫ 2 ( ) is ǫ ∗ j δ δ a Λ i ) m B τ p δ – 8 – ˙ ˆ 3.8 αi dτ E N := := − provided ǫ a , so that one sees that the relation ( a α Z = dτ 2 = 2 ˆ ˆ = (¯ E E 2 dτp ǫ m − ǫ m b kj Z i i ) p αj i Z ] 2 κ = = ik = 2 ij τ ¯ p ij − 42 -symmetry with ( = ) = a ˆ ¯ ij p a E κ p ˙ p αi 2¯ p S = a ) is also related to the wish to have the ij ¯ ǫ ǫ a ij p p , δ ¯ − p p S ǫ 3.1 2 α i + ] to the study of different, more conventional models. In the n δ / ǫ ˙ αα ij a τ 42 N ˆ ˜ σ E = ( ˙ αi ij = α : ) are the parameters of the fermionic variations defined by p ¯ κ ǫ ˙ a αi ik + ¯ p ¯ ǫ p 32) , a | τ ik = ˆ α i p E ǫ 0 (60 α i a we will present an improved superparticle model which does n θ p Σ ( κ = ( δ dτ ) and the variation of the action ( 3.2 α ǫ Z In the case of flat central charge superspace Σ De and Azcárraga Lukierski appreciated themselves that im Then the fermionic variation of the action ( 2.19 = S in ( The above de modelaction Azcárraga-Lukierski inspired us to stud observation, passed in [ section constraint by hand to possess 3.2 Improved superparticle model in central charge superspac parameters of the action ( by hand at this stagefind does it not similar look to consistent imposing (although the in section th condition in EFT) and, were restricted by the condition This implies ¯ It is not difficult to check that these obey the constraints would be a gauge symmetry of the action where Now one can easily check that [ JHEP01(2016)132 ) are ) is 4.1 tion ), (4.2) (4.1) e ). (3.15) (3.16) (3.17) (3.18) roduce are the 3.14 3.8 3.13 and ij τ j . ˆ ¯ i E . ) kj ) results in the ¯ p ˙ ). αi ij erparticle model, ¯ ik and κ ˆ , , ∂ ˙ 3.6 E αi ij τ ij ¯ θ − ˙ ¯ ˆ ). The Lagrange mul- αj ∂ τ E a ( ¯ κ ∂ = , ); finally Λ , and then quantize this, cks of the Cartan forms ij a ep τ r + 7→ e described by the second p htforward quantization of ˆ ionic variation of ( kj ) and ( p E 2 i α ) = ˆ ¯ 2.20 E κ ij a τ − ik α i laced by particle momenta so ˆ on involving the Sp(56) met- 3.13 E , p i α θ ( . κ τ k ij ] ) supplemented by the Lagrange p ∂ j ) and, using these one can deduce ˙ ( αα ¯ p i ) and ( a k = 0 2 , p i ˜ , σ [ 3.12 ′ a 3.13 l ij − ′ p 2.19 k ˆ E p = Λ = 0 ′ = = 2 , we arrive at the general solution of the j 4 r ′ kl i m ˙ ∂ αi p ij τ p ¯ ) and ( θ ′ ˆ ¯ l kl E = ′ ¯ 7→ p k – 9 – 3.6 ′ j kl j i r ′ ˆ ¯ -symmetry ( δ p E κ ), the SU(8) part of which coincides with ( jl 1 8 , p ijkli , δe k ǫ ] + are independent variables, 1.7 ik ) j ) by 1 4! , δ p ˙ αi kj i ij ij [ p ¯ κ p k equations give the standard − αj ) only. However, if we first find the general solution of ( , ∂ ik ] ˙ κ αj a ¯ p ij ). ¯ ) by a simple prescription (¯ kl θ p , p ij ¯ 1.5 ∂ ¯ τ SU(8) part reads p = Λ ) in terms of smaller set of momenta p and ¯ ( / ∂ ij ij 2¯ 2.1 4.2 ij [ ij τ ˆ ¯ E ¯ ij p + ˆ 7→ − , p E p 2 j α 4 ) and, in the case of generic curved target superspace, by ( ) ij 7(+7) , m ˙ κ p αα a ) provides us with a counterpart of SU(8) part of the set of sec ij ) (while E a p α i ) and ( = ˜ σ , p θ 3.11 3.5 τ ˙ ij ). Indeed, the equations of motion for the internal momenta p αi 3.13 kl 4.1 ∂ p ¯ κ )) of the flat supervilebien forms ( ˆ ( E a i ). To be precise, if section conditions appeared from the sup 3.1 jl p 3.3 ¯ p 2.8 = = 4 ik ) the following interesting interrelation between pull-ba ¯ p j α i i ) expressing (¯ ), is obeyed identically. Λ δθ 3.17 δ 4.2 2.9 As we have already discussed in the Introduction, the straig The action is invariant under the Notice that this model is not apparently equivalent to the on However, despite the lack of apparent equivalence, the ferm strong section conditions ( tiplier variations produce the constraints ( supplementing (¯ The classical mechanics counterpartric, ( of the section conditi the weak section conditions ( the constraints ( and ( 4 Superparticle and sectionNotice conditions that eq. ( still described by ( from ( the relations pull-backs (( multiplier transformations: Here the momenta Lagrange multipliers introduced to impose the constraints while the remaining which differ from ( that we would have the constraint ( conditions ( in the classical approximation the derivatives would be rep order action ( JHEP01(2016)132 ] 31 , (4.4) (4.3) (4.5) SU(8) 30 / ) of the )). First symmetry 4.1 constraints 7(+7) 3.13 7(+7) E E = representation of = 8 supergravity ], the flat superspace 30 N 28 ) which is preserved ) is dependent, i.e. is G/H nontrivial U(8) part ( + kl ) constraint only on the Σ group; the 4.2 28 nder the SU(8) subgroup oing to show that this is ather a speculation), one , δ = 8 supergravity [ . ] is realized dynamically in enerates the SU(8) part of 4.1 = y invariant under nontrivial iew it looks natural that it kl : 7 7(+7) 32 f ( ∗ Σ N 56 undamental representation of volving the SU(8) generator. ) E δ ions coincide with the general I the superparticle model, which p n of the classical section condi- the flat superspace limit. ,..., = ( symmetry of s of these, do not see any effect of the = ( ) (or equivalently, eq. ( 7(+7) = SU(8) and = 1 I (Π) ). In other words, it inspires to check E p 4.1 H Σ 7(+7) 4.1 V s the derivative of constant matrices vanish and E , ∈ 4 ,I / ) 2 ij symmetry is realized trivially in flat superspace limit. = 8 supergravity [ SU(8). Indeed, if this were the case, it would kl p / m SU(8) transformations. This implies that in flat – 10 – Σ N = = ¯ ⊗ V 7(+7) , I ij 7(+7) E ) p kl 8 E I Σ ˜ γ p = SU(8) part ( 7(+7) can take any constant value in I V = E γ H ( symmetry, may generate additional (Π) I = ( should obey Σ p I V ) are the SO(8) Clebsch-Gordan coefficients. If we wont to symmetry of G/H p ˙ (Π) = q Cartan forms, influence the superspace geometry [ 8 p Σ 7(+7) ij V ¯ , γ p E ˙ q 7(+7) ) may reflect the fact that their G/H part ( I 7(+7) p γ E E ) with symmetry generators of 1.7 -valued matrix ‘bridge’ superfield 5 = ( 4.2 ˙ ˇ I qp 7(+7) γ E ), the seven real = ˜ ) and ( ‘survives’. Namely, the superspace formulation of ˙ q ˇ and by left multiplication on a matrix in a reducible I p 4.1 3.13 SU(8) transformations. γ / 7(7) Let us begin by discussing the solution of eq. ( Then the superparticle model in flat superspace is manifestl The superparticle model discussed in the previous section g The above discussion suggests that the appearance of only a S To be more precise, one can state that E 5 7(+7) 7(+7) is realized by multiplication ononly these these, matrices. through However, the a solve ( by SU(8) subgroup of the above Indeed, in this limit the matrices which is transformed by rightE multiplication on a matrix in f superspace limit the (rigid nonlinearly realized) In it SU(8). In flat superspace limit one usually sets makes tempting to search the explanation of the appearance o coordinates, as wellE as any dynamical system described in term whether the general solutionsolution of of all the the set SU(8)indeed of the part section case. of condit the section conditions.notice We that are it g has a solution with 7 real parameters section conditions ( satisfied as a consequence of its transformations of SU(8) symmetry,generates and, as from this constraints point onlyHowever, of a accepting v part this of argumentcannot (which section conclude that is condition a not in curvedis superspace a generalization expected proof, of to but see r the of involves an basis of the fact that such a way that in its flat superspace limit only the symmetry u reduces to SU(8). corresponding to the coset contradict the physical requirement of non-singularity of the classical section conditionstions only. ( Then the associatio JHEP01(2016)132 , e pq ) and ives 8 (4.6) (4.7) (4.9) (4.8) I pq ˜ γ (4.10) ons of I ij (1) γ p and hence = ( I ) is proved to I pq ) up to SU(8) = 4.4 † 4.1 . UU ] ′ = 7 gamma matrices s ′ r . d ) 8 f this solution-generating ˜ ! γ ) of the whole set of section i ) ) implies that the matrices i J | , − ) for n coefficients, Γ ons of this equation, ditions, and ( 4.9 γ ents should hold 7 4.1 ( tion conditions found by Hohm . 1 1 ′ 4.8 q ] ′ which obeys ′ nce of a part of section conditions. p )

s , [ ′ J ) shows that it is equivalent to the ) | r ,..., 8 ⊗ jk (2) Γ ˜ γ p ′ 4.4 | q = 0 ij pq | I ′ = 1 ik ( I γ p ( [ pq γ it reads 2 (1) Γ ˜ ( q ¯ ′ ) solves also the G/H part of the classical ′ p 1 √ s s I pq ′ ′ 4 4 m r r ) when written in the SU(8) basis. Then, as ′ ′ 4.4 q q ,I ′ ′ ) and thus provides a solution of the complete = = , 4.4 – 11 – ij j ) describes general solution of ( S I i 4.2 p p ]) ). ] pqrsp pqrsp U q 8 ǫ ǫ 4.4 = 43 δ ], work in the ‘SU(8) basis’. The relation between 1 1 4.2 p 4! 4! I ij [ I , δ Γ 32 = = I ). ! p ] ] ) = 2 pq pq ) can be written in terms of SO(7) gamma matrices Γ J 4.2 rs rs | ˜ = q q ) and ( ) pq Γ 8

| 4.4 ij q ˜ I γ ). This is not manifest from the ﬁrst glance as for this to be th ¯ pq 4.1 p ( [ ) S ]. This was written ‘in SL(7) basis’ and reads J Γ | 4.2 = γ ) and SU(8) decompositions of the fundamental representati 19 ), as only SU(8) part of U(8) transformations of the momenta l ( , R EFT coincides with ( ! , pq 4.1 [ ij ij 17 ) ¯ p p 8 ˜ γ

| 7(+7) I ( E are not degenerate, so is γ ( ij (2) p ) for the SO(8) Clebsch-Gordan coefficients. can be described by (see e.g. [ 4.7 and At this stage one can wonder whether ( Now we are going to show that ( In this paper we, following [ Notice that, by pass, we have proved the identities ( However, a more careful study of the solution ( Actually, the solution ( , are related by an U(8) transformations. Indeed, as ( 7(+7) ij (1) ij (2) has been used above. E the representation of which in terms of SO(8) Clebsch-Gorda As the condition for SO(7) gamma matrices Γ belongs to U(8). BelowU(8) we symmetry will of concentrate ( on the SU(8) part o and ( p set of section conditions, ( transformations. To this end, we first notice that two soluti conditions of Hohm-Samtleben solution solves the wholebe set equivalent of to section it, con this solves also eq. ( components of SL(8 classical counterpart of the solutionand to Samtleben the in whole [ set of sec section conditions, eq. ( case the following identity for SO(8) Klebsh-Gordan coeffici p Using this we find that the solution of Hohm and Samtleben ( but this is not essential for the conclusion on interdepende invariant also the constraint ( JHEP01(2016)132 . . . I I p w can ). Of = 7(+7) r fact ij (4.11) (4.12) I p E 2.8 w ), rmations, = ¯ plex ¯ I 3.13 ]. Namely, we w ) and ( 24 – 2.7 17 is real, eneral solution of the I with some complex ich constitute the gauge w ng of the structure of the I ij ection conditions for other s issue in the future. s sufficient to describe the Γ ) contain 14 real parameters ), dulo SU(8) transformations’ fields on not more than 7 of I to SU(8) subgroup of tion conditions which involves w n restricting the dependence of lds on additional coordinates is 4.4 4.11 n conditions ( = ¯ ries (EFTs) [ , ij p I ij , Γ I ij w effective field theory it is not necessary to ∂ = ¯ = EFT. ij I ¯ p 7(+7) ∂ ]), the ‘quantized’ version of eq. ( ). To prove this, let us show that an arbitrary E ij . ) only in the case when – 12 – ) ) and this latter is SU(8) invariant, we conclude 30 7(+7) 8 ) under SU(8) is generated by 42 elelments of the 4.4 E ˜ 4.1 γ , 4.2 ) can be obtained by SU(8) transformation applied I 7(+7) γ ij I ij 4.11 E Γ ). , p I = ( ij w 4.4 p ij ) of the ¯ ∂ = = (¯ 1.7 ) provides us with the general solution of the complete set of ij EFT it is not necessary to impose the complete set of section p Σ p 4.4 ) solves also ( ) starting from ( = 4 supergravity [ 4.4 7(+7) ) is SU(8) invariant, we conclude that, modulo SU(8) transfo D 4.1 E 4.1 ) and are invariant under SO(7) subgroup of SU(8). This latte solves the constraint ( I . = 8 I I ij w w Γ N I SO(7). Finally, as 42 + 14 = 56, we conclude that arbitrary com / w = ¯ exceptional field theories are also reducible. ij ) p n To this end, let us firstly notice that the expressions in ( It will be interesting to understand whether the sets of the s To conclude we have shown that in the Furthermore, as ( Interestingly, just the freedom in SU(8) transformations i This result implies that, modulo SU(8) transformations (wh (+ n On the way westrong have section also conditions proved implies that,56 the as additional dependence might bosonic of be coordinate. the EFT expected, the g be generated by SU(8) transformations starting from ¯ Now, ¯ E 5 Conclusion and outlook In this paper wemysterious have section made conditions a of stage the towards exceptional better field theo understandi with complex to one of the vectors described by the complexification of ( As the constraint ( implies that the nontrivial orbit of ( have shown that in course, it looks desirable tomore make concrete the in above the statement quantized on version. ‘mo We hope to turn to thi that the SU(8) orbitclassical of section ( conditions ( its general solution is given by ( (in 7 complex coset SU(8) general solution of ( nondegenerate 56-vector conditions as all the jobdone on by restricting the the part dependence of of the fie section conditions which corresponds impose the complete set offields section on conditions as additional all coordinates the isthe job done o generator by the of part SU(8) of subgroup the of sec symmetry of provides the general solution for the complete set of sectio JHEP01(2016)132 ]. = )) 42 N ) can 3.13 ). The 3.12 EFT when 2.17 4, see ( / )–( 2 7(+7) m E j 2.12 i EFT in curved δ − e used in our study is he superparticle model ) of the classical section = ], while the quantization 7(+7) e conditions for external lly interesting question is 4.4 orsion ( E or superfield formulation of 44 kj ate superparticle models in ersation, to Paul Howe for ng section condition can be oposed in section 2 of [ p inking suggests to relate this model (and which should be ic counterpart of the massive 11 dimensional superparticle. n EFT, is the property of M- general solution of the strong ] for such approach to D=10 e these as their selfconsistency ralization of our superparticle ik ns and collaboration on early EFTs are also reducible. ever there exists an alternative vitably result in appearance of p icle is massive (in 4-dimensional 48 f superfield formulations of the ) erparticle model. In essence, the hanic counterparts of the section – -symmetry, similar to ( ution ( ]) related the square of external n uld describe the κ ts. 45 (+ 42 on of the n E = 8 superspace enlarged by 56 central N ), under = 4 2.11 = 8. In particular, this might help to understand – 13 – . The basis of such a formulation can be provided D n 32) | = 8 superparticle model [ (60 N = 4 6 and with D ≤ . Hence we can treat the superparticle as being massless in an 6= 0) might look strange as to obtain supergravity one would 2 n m 2 6= 0) and of the internal momenta (¯ m 2 m = a = p a a p p EFTs, with ) with generic supervielbein ( a ) p n (+ 3.14 n E = 4 central charge superspace Σ One of the natural applications of our approach is in search f Another interesting direction for future study is to formul The superparticle model in central charge superspace we hav This progress is reached using the suggestions coming from t D by an appropriate set of the constraints on the generalized t 8 exceptional field theories, particularly for the formulati stages of this project, to Henning Samtleben for useful conv Acknowledgments The author is grateful to for Ort´ın Tomás useful discussio rather quantize a massless whether the sets of the section conditions for other enlarges spacetime. In particular, on the 7-parametric sol central charge superspaces appropriateother for construction o whether these still hypothetical constraints, if found, wo requirement of the invariance ofmodel, the ( curved superspace gene allow us to findand a D=11 set supergravity of in appropriate constraints standard (see superspaces). [ An especia conditions the model should become equivalent to a massless modes of String theory.and internal Furthermore, momenta, which onepartially follows can from imposed observe our by that superparticle momenta hand th ( in theto original the model same of parameter [ of massive superparticle withhigher 32 spin supersymmetries fields shall in ine theto spectrum. the However, fact a that more U-duality,theory careful which and th is higher expected spin to fields be should appear manifest here i as M-theoret section conditions starting from its classical counterpar This is a massivecharge superparticle coordinates. model On in first glance,perspective, the fact that superpart in central charge superspace.reproduced We starting have from discussed the how classical‘first the constraints quantize of stro than a solve’ sup way ofconditions treating reproduce the the classical weak mec section‘first conditions solve then only, how quantize’ way which allows to reproduce the supplemented by the section conditions orconditions. they would These produc issues are under investigation. an improved version of the de model Azcárraga-Lukierski pr JHEP01(2016)132 of ] ] , ommons (1995) 109 , ]. ]. 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