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E 8 in N= 8 in four

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Citation for the original published paper (version of record): Ananth, S., Brink, L., Majumdar, S. (2018) E 8 in N= 8 supergravity in four dimensions Journal of High Energy , 2018(1) http://dx.doi.org/10.1007/JHEP01(2018)024

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(article starts on next page) JHEP01(2018)024 = 4. Springer d January 8, 2018 ck to : December 23, 2017 Dimensions November 30, 2017 : : Published , Received Accepted = 3, a field redefinition to d a and Mathematical Sciences, Published for SISSA by https://doi.org/10.1007/JHEP01(2018)024 invariance. Our procedure to demonstrate = 8 theory to 7(7) N E [email protected] , and Sucheta Majumdar b,c = 8 supergravity in four dimensions exhibits an exceptional N . supergravity in four dimensions 3 Lars Brink a 1711.09110 invariance manifest, followed by dimensional oxidation ba The Authors. = 8 c Supergravity Models, , Theories in Lower

8(8) We argue that , E [email protected] N in symmetry, enhanced from the known 8 8(8) 637371, Singapore E-mail: Pune 411008, India Department of Physics, ChalmersS-41296 University Sweden G¨oteborg, of Technology, Division of Physics andNanyang Applied Technological Physics, University, School of Physical Indian Institute of Science Education and Research, [email protected] b c a Open Access Article funded by SCOAP Abstract: Sudarshan Ananth, E ArXiv ePrint: E Keywords: this involves dimensional reduction of the render the JHEP01(2018)024 4 6 7 7 9 1 2 5 5 7 8 10 11 12 = 4, d ] of the 4 mal super- ]. This theory is known 1 has recently been shown maximal SU(8) this has been achieved up = 8 supergravity, exhibits e an underlying enhanced ant. Since the the states of N maximal supergravity, in symmetry in eleven dimensions. symmetry symmetry [ 7(7) E 8(8) 7(7) ], we showed that “oxidation” [ E E 3 – 1 – = 3 — version I = 3 — version II ]. There is mounting evidence from these calculations d d 2 = 4 preserving the d = 11, suggests that there is an d = 4) supergravity in light-cone symmetry symmetry , d 7(7) 8(8) E E = 8 and the exceptional = 8 invariance = 11 theory are not representations of the linearly realized N 8 N d gravity 5.1 The field5.2 redefinition SO(16) symmetry revisited 2.1 4.1 E = 8 theory to is constructed as ato power the series four-point in coupling. the In coupling an constant earlier and paper [ N to be betterto behaved in be the finite ultravioletand up than others to pure that four gravitysymmetry. points loops and Using to light-cone superspace, [ unexpected the Hamiltonian of cancellations and henc Gravity with maximal supersymmetryboth in four dimensions, 1 Introduction B SO(16)-invariance of the new Lagrangian C 6 Oxidation back to 7 Conclusions A Verification of field redefinition 5 Relating the two different versions of three-dimensional maxi 3 Maximal supergravity in 4 Maximal supergravity in Contents 1 Introduction 2 ( This result has beenthe shown to first order in the coupling const JHEP01(2018)024 8(8) (2.1) E e SO(16) to see the invariance to 8(8) E , ∗ of SU(8) respectively. ) riting the Hamiltonian ¯ s involving dimensional x 8 der in the gravitational y have unique quantum ies discussed earlier, thus s we can make particular ls theories have as many heory. This means that we = 8 superspace is spanned = ( to find all the symmetries. n all dimensions (note that and = 4) supergravity in light- ur-dimensional one but with x symmetry, at least to second N at leads us to conclude that 16) invariance and a full eld redefinition that maps this 8 ked with symmetries, some of heory with onal . Three such states mensional theory without one. e” the second formulation back rits a three-point coupling and , d s, since many of the symmetries ) ¯ 2 8(8) E = 8 i x . The ∂ . + 8 ), the N = 4 theory in a straightforward way to 1 x 7(7) d ... and ( E 2 ¯ , SO(16), in a linear fashion since under this ∂ 1 , √ = 1 ∓ – 2 – 8(8) ∂ = m E ( x m ¯ θ ) 3 x and ± 0 m x θ ( +), the light-cone coordinates are = 4 exhibits an exceptional 2 , 1 d √ + , = + , ± − x = 4) supergravity in light-cone superspace , we review the formulation of ( 2 , d symmetry they have to be broken up into such representations ]. We can then go back to the first formulation and indeed find th 5 = 8 7(7) E N In section Motivated by this, we describe in this paper an entire proces Our formulation uses only the real degrees of freedom of the t = 8 supergravity in by the Grassmann variables order in the coupling constant, enhanced from with the corresponding derivatives being symmetry [ This form of the three-dimensional theory exhibits both SO( symmetry, the states of the theorycannot transform form as a 128-dimensi scalar. Inthree-dimensional the theory, following with section, we a present cubic a fi vertex to a three-di arrive at an action,only in one three dimensions, transverse derivative. whichcannot mimics This exhibit the formulation the fo thus maximal inhe subgroup of reduction, field redefinitionsN and dimensional oxidation th cone superspace. We dimensionally reduce this symmetry there, now realized non-linearly.to We then four “oxidiz dimensions inarriving a at manner a that four-dimensional preserves maximally supersymmetric all t the symmetr symmetry. This is accomplishedthe by using number the of same states superfieldin is i this always 128 formulation bosons wecoupling + can constant. 128 prove fermions.) the symmetry By w to the lowest or of the that order. 2 ( symmetries visible but oneWe formulation strongly will believe neversymmetries that be as enough maximal one supergravity canproperties. and pack into Yang-Mil one theory, and this is why the lose a lot of theare covariance usually non-linearly found realized. in gravitywhich In theorie are a sense difficult the to formulation see. is pac By making various field redefinition With the metric ( JHEP01(2018)024 tor (2.5) (2.6) (2.7) (2.8) (2.3) (2.2) (2.4) ] . 6 8 . . . d , ) 2 , d y ) ], ( 1 , y 7 the 28 gauge fields, ( d + ¯ φ , ∂ mn ≡ mn mnpq 8 , 4 n ¯ ¯ ) ¯ A ¯ ) A θ 8 C + , ) d y q . ∂ 2 ( ( + ) , d i θ 1 y ( ∂ ) ¯ h √ 1 4 p ( d by the superfield [  y 2 θ n rators are [ . ( u m − corrections order by order thus θ + n , ¯ = θ n ψ , θ tu ∂ n m + + ) m + ¯ q A θ ∂ . m ∂ y θ ∂ ( gravitinos, θ ∂ θ i 2 + n + ¯ 2 ∂ ∂ + ¯ θ 2 3 ∂ i ∂ , φ 1 4! ∂ ∂ = stu √ 2 i χ ) + n m = n mnpqrstu √ − y δ + − ǫ ( n 2 q ) u mnpqrstu + − − satisfy ) = 0 y m ǫ θ ¯ √ q ( x ¯ i n t t y ψ ¯ mnpqrstu φ ǫ θ θ ( ∂ 2 ≡ = – 3 – , s s s mnpqrstu the 8 spin- ; ¯ ¯ ∂ θ φ + 1 mnp ǫ θ θ θ } − + + m ¯ ∂ n χ r r r r m n ¯ = ∂ q d ¯ θ θ θ θ − ψ , y m + n ¯ q q q q 1 q m + ¯ + ¯ ∂ ∂ d , θ θ θ θ θ ∂ i θ p p p p p 2 m − θ θ θ θ θ i q = ; ¯ , while the dynamical ones are given by x, x the 70 real scalars. These fields are local in the coordinates √ n n n n n { ) + + θ θ θ θ θ + m y − x, ∂ + ( ∂ q m m m m m m n h  θ θ θ θ θ m m mnpq δ ) = 0 ; ¯ 2 θ θ ¯ ∂ 2 i i C 1 1 4 = 3! 5! 6! 7! 8! y + 1 = 8 supersymmetry algebra closing to the Hamiltonian genera ∂ 2 √ ( i y ∂ i + + + − + √ φ − N = m = − } d ) = n + m and its complex conjugate m y + q ¯ θ ¯ ∂ q ( φ represent the , , ∂ φ ¯ h m + − q { = and m h the 56 gauginos and d mnp ¯ The kinematical, spectrum generating, supersymmetry gene satisfying These satisfy the free In the interacting theory, thegenerating dynamical generators the pick interacting up Hamiltonian. The superfield χ All 256 physical degrees of freedom in the theory are capture where where JHEP01(2018)024 , for- ) (2.9) 2 (2.11) (2.12) (2.10) (2.13) (2.15) (2.14) κ ( , O , symmetry,  n φ + , a constant. p . + T 7(7) = 1. ∂  =0 q pqrs η E 8 pq

) Ξ m d c.c. θ  δ . ( φ ] φ + 9 θ − + q 8 +3 q ∂ ∂ d ¯ this order is ∂ ∂η ∂φ -model symmetry of the m , ˆ ¯ d R mn p + σ ) klmnpqrs η T d is no longer shown. ǫ ∂ − n 1 2 ∂η ¯ ar ∂φ p ransformations ¯ ∂φ∂ δ n + φ e + 3 = + ∂ ormalism, the h the superPoincar´ealgebra as physical fields in the supermul- mmutation relations with other ∂η φ +3 φ∂ ] = ¯ φ∂ ∂ m m ˆ ¯ are given by [ q +2 d 4 ¯ , q , klmn ∂ p η ] ∂ + 1 ) κ e + 8 q Ξ ∂η L ∂  d ∂ − ,T ¯ θ ∂ φ 8 n − ≡ − +2 1 and + m φ φ ∂ θ d ∂ 2 n 2 mnp T 8 + d d + m − mnpq ¯ d 4  ¯ Z φ∂ ∂ mnpq – 4 – ;[ ¯ ∂φ∂ − x m  ∂ ∂ .... ¯  ¯ ∂η ∂ 4 q p 1 ¯ φ ¯ d φ ∂ ∂ ¯  q ∂η ( m 4 p ¯ +3 d Z  q + 1 + 1 = 2 ( ∂ ∂ n β = φ ∂ m  and ✷ n = + δ 1 mnpq κ ∂ , φ 1 8 Ξ 3 4 ) = 8 supergravity reads [ ...m + 1 κ m κ ( ¯ 4! algebra. In compact coherent state-like notation the trans + − ∂ d m N m mnpq d φ m n + d − ¯ , q η ¯ q κ 7(7) n  m +4 ✷ θ SU(8) transformations to order = E q ∂ / m ˆ ¯ +2 1 mnpq one leading us to question which is the more basic one. d  ¯ klmn θ φ ∂ η l Ξ Ξ + θ 7(7) − k ∂ and 7(7) E θ i = 2 E mnpq 1 mnpq 64 klmn and Grassmann integration is normalized such that ) can be written √ = θ Ξ L θ symmetry 2 − , the action for 2 κ κ 2 κ 4! κ = = 2.13 π G − klmn 7(7) + − 8 n β θ E = m √ = The correction to the dynamical supersymmetry generator at T δφ = δφ κ where 2.1 where the + sign on the kinematic supersymmetery generators The non-linear These 70 coset transformations along with the linear SU(8) t The d’Alembertian is generators like the superPoincar´eones. Note that in this f This formulation is particularly useful for checking the co mation ( To order The action to order where where which is a duality symmetry of the vector fields and a non-line constitute the entire scalar fields in the covarianttiplet. formalism, Hence transforms all the the supermultipletwell is as a of the representation of bot JHEP01(2018)024 3 ]. φ ly κ 10 , (3.2) (4.1) (3.1) (4.3) κ invariant 8(8) , E  endix A). Before ) representation c.c. = 3 theory (up to an + 16 ) (4.2) d . Maximal supergravity . y 3 ∂φ 8 ( on to the e dimensions. We discuss + φ + m ¯ supergravity q 8 ∂φ∂ was extensively studied in [ , since three representa- 0 , form a m + = ǫ 8 bosons and 128 fermions, φ 1 m being the parameter. ¯ θ n section ¯ φ∂ = 4) ǫ + 16 4 , ) = f 1 = 3, we are left with the dependence y + 1 , d L ( generates the 120 SO(16) transforma- and ∂ d ¯ θ 28 φ 8 128 φ U(1) as follows. m , = 8 − + symmetry. The same chiral superfield θ kin θd s + × 1 φ 8 N 2 b − ( + 8(8) xd U(1) , δ 3 28 E – 5 – ) d 128 φ∂ × y 2 + ) ( = Z 0 φ ¯ φ∂ = 3 — version II = 3 — version I = 4 theory to = m 4 63 generaors on q d d + 1 d 256 S SU(8) m = ∂ m ǫ q on the superfield  ⊃ , ¯ . We obtain, for the action for the κ m 120 m ∂ ) = ¯ 4 3 q q y , ( + -symmetry. The action of the linear SO(16) and its non-linear φ = 3. This theory does not admit vertices of odd order ( m q φ R d 4 kin ¯ SO(16) s SO(16) on the light-cone superfield -invariant version δ + / ✷ 8 ∂ E ¯ φ 8(8) − E = here, is the three-dimensional d’Alembertian (see also app L ✷ = 8 superspace, the Grassmann variables, the manifestly obtained by dimensional reduction from ( ( The linear action of ¯ N supergravity theory in we study the symmetries of this action, we divert our attenti etc.), due to the SO(16) realised quotient where the SO(16) invariance of the theory In yield the kinematical light-cone , with There is a betterthis known version in form this for section, before maximal relating supergravity it in to the thre form i When we dimensionally reduce the The action for this theorytions contains cannot no form three-point a coupling scalar. on one transverse derivative, The quadratic action of the where in three dimensions is invariant under an 4 Maximal supergravity in overall constant) tions, which are decomposed in terms of SU(8) 3 Maximal supergravity in introduced earlier describes all the degrees of freedom: 12 JHEP01(2018)024 ] ′ 1 , 10 ) 28 ) (4.6) (4.8) (4.7) (4.4) (4.5) 2 T. ) κ SO(16) ( orm [ p / of SU(8) U(1) rep- O m δ 8(8) ] 28 × + q n E of the SO(16) 10 δ =0 η and

− 28 ϕ ,  ) + q 28 n − φ ¯ m q + ) y δ c and ( ∂ m p − ¯ q n . δ , 28 +(3 mn . n . mnpq ¯ ∂ q perfield contain a constant α T ϕ ,  ˆ ¯ 2 ensional representation d β · m ′ − ] = 0 F η ¯ = = q + 2 ( +2) ¯ 1 c − n c + p ϕ ϕ + 1 , m 2( + ∂ mnpq n 2 of the bosonic fields, and ∂ ) φ e ˆ d θ ) 28 1 1 2 T − − c q ′ − U(1) , +2) y − − c ··· ( = 1 SO(16) in terms of SU(8) m T,T ) 2( 2 / δ 28 ¯ [ β +(3 − − m m -model. SU(8); the rest of the coset 2 mn + + y ∂ ˆ d 0 ··· σ / 8(8) ( + ˆ ¯ 1 2 , p d 0 · 1 E n , m m η , ] 1 e θ ∂ 7(7) e ˆ d mn 70 T U(1)) are generated by the q m  β E m – 6 – ˆ d + ¯ ,T q m × = 2 ≡ 2) +  δ 1 ′ 1 , n c − ) and U(1) generators are given by [ ∂ 2 q ) + 4 c m − − 28 2 − m +2) q +( = c X q y mn [ q c 8 ∂ 2.14 ( + 2( n ϕ , δ + (SU(8) m 1 8 + i θ ′ 2 ϕ , δ m / ∂ T n , (which are not related to the ∂ ··· 1 mn n p ··· q 2 1 1 + symmetry can be represented on the same supermultiplet i 2 2 ...m β +5 m m − 2 ∂ m = c ) + ′ m δ √ 2 + q = T 1 m − 4 1 ∂ m 8(8) are the transformation parameters. m y 28 m − ( m ˆ d  n E mn mn 128 = q mn U(1)) β ω α T δ mn κ ǫ m T δ η 2 e θ × α i = = T + 1 SO(16) coset transformations can be expressed in a compact f +  p / ∂ + ϕ ϕ n ( F as the representation in 8 δ SO(16) variation, just as in a × = 1 28 κ 8(8) / , and δ SU 70 E = δ F mn 8(8) ] = φ symmetry. α E symmetry pq , m 7(7) ,T n 8(8) SO(16) E ω / E mn All the 128 T 8(8) [ E δ and its complex conjugate transformations form two U(1) singlets, a twenty-eight dim 4.1 We decompose the non-linearly realized coset resentations as the where the sum is over the U(1) charges where discussed previously). Allterm the in the bosonic components of the su We identify the It is remarkable that the and The coset transformations SO(16) The SU(8) generators are given in ( which close on (SU(8) Hence, the linear SO(16) transformations read JHEP01(2018)024 rom (5.5) (5.2) (5.1) (5.4) (5.3) = 4) . We ) piece ′ , d ¯ φ ′ φ , = 3 action = 8 obey aximal su- d  N ). The SO(16) c.c. in the interaction , 4.6 , + ) ′ 2 ′ + ¯ ∂ φ  ∂ ′ F ∂φ ¯ φ relate the ) to a kinetic term plus + + + by ∂ ∂ nsionally reduced form is ∂ ′ ′ ′ 5.1 − φ φ . non-linearly realized on the vity in three dimensions, we ∂ ∂φ E 3 ld redefinition. + + + 1 is listed in ( ∂ ∂ ∂ ′ = 2 ( ) = 0 (5.6) ¯ φ φ  . D F 4 4 δφ ) + + 1 ( 2 + + ∂ 4 κ ( + E κ∂ ✷ βκ∂ − ∂ O 3 2 ′ ) as shown in appendix A. The ( + ¯ φ φ + ) + 2 ′ − 3.2 ) + φ + ′ φ at lowest order. Thus the the dimensionally 4 φ ∂ φ C ′ ,D 4 + – 7 – ✷ + φ + φ ∂ + ✷ 2 ∂ ∂ ′ ¯ ′ ∂ φ ∂ = ′ = 2 ) φ φ − ¯ ¯ ′ φ φ + B C 4 δ φ ∂ + = ( ( + 1 ∂ + κ − ∂ ( ) at the free order implies L ), obtained from dimensionally reducing ( 3 1 A B  = + 5.1 κ + 3.2 + L ′ 3 4 δ ), since φ A invariant supergravity theory sans a three-point coupling ακ∂ + 3.2 = ′ + φ ′ φ ). Based on dimensional analysis, we start with the ansatz for simplicity.) 8(8) 4 φ = 3 maximal supergravity with a cubic vertex can be obtained f + E ✷ → 3.2 ∂ δφ d = ′ φ ¯ φ φ ≡ − symmetry revisited φ = are constants to be determined and the A, B, C, D, E, F ′ L β SO(16) , ) terms in ( δ κ α SO(16) ( pergravity O reduced action for The Lagrangian for the SO(16) invariant theory reads will do this throughalso a invariant field under redefinition SO(16) andsuperfield. transformations, show which that are the now dime 5.1 The field redefinition terms. We thus arrive at the new Lagrangian which exactly matches ( where invariance of the Lagrangian ( the supergravity to the We want a field redefintion that will map the kinetic term in ( (Note: 5 Relating the two different versions of three-dimensional m in the field redefinition achieves the same effect as replacing with a three-point coupling ( Having described the two differentare forms now of in maximal supergra a position to establish a link between them. We will which correctly reproduces the cubic terms in ( Simple computations lead us to The linear action of the various SO(16) generators on the SO(16)-invariant action (without a cubic vertex) by5.2 a fie JHEP01(2018)024 8 E (6.3) (5.7) (6.2) (6.1) . =0  η ) = ¯ φ , b ) to obtain δ Lagrangian = ( ) a .

+ 5.4 obtained by di- =0  = 3 η =4 1 φ∂

d d 3 − , i  + 1 ) φ m ∂ ) φE c , φ dynamical supersym- W symmetry.  . − m 3 2 4 +5) ¯ (4 d , c κ an obtain the light-cone + 1 + + m ( , ˆ ¯ + , we invert ( d ¯ ∂ 8(8) ∂ d ′ + η m ∂ κ∂ ¯ φ E q 1 m ) by a generalized derivative m t this new theory is also − , 1 3 2 − η ˆ ¯ q symmetry E∂ ∂ + W bǫ an  − x ) is also SO(16) invariant albeit = φE 3 , ∂ − ) (=  ˆ c ˆ ¯ ∂ 8(8) d +2) ∇ ¯ ¯ ∂ θd a 5.5 φ c − ( 8 E − + 2( φ (4 ] e ∂ , θd + , ) , η 8 11 ...m ≡ 2 dyn s d 2  + m δφ δ 1 E∂ ¯ ¯ q φ ∂ i∂ m ( h − 1 Z  3 + m c symmetry. We achieve this by introducing a + m E + 1 2 2 1 i φ∂ b ǫ  ∂ + ∂ √ 2 3 −→ – 8 – ∂  δ 8(8) 4 δη + 1 = supergravity, is equivalent to the 8 4 and E ∂  ˆ ¯ + q =3 ≡ ∇ ≡ ˆ ¯  d ...m d δ i ) η δb 4 κ∂ ) = 4) +5 + c m + 3 2 ˆ ¯ q δ 2 δa , φ , b ǫ , d bǫ = 3 Lagrangian, without cubic vertices, may be oxidized W m − κ∂ m , ( + = 4 by replacing all the + ¯  ) 2 3 d ˆ d ∂ ∂ . m ∂ = 4 preserving the φ , a = 8 d 2 δ e δη − a +4) + ∂ m W = 3 c d 1 ¯ ) ( q ( ∂ N  ≡ ) , d ( φ = + 2 8 + + ∂ )! E 1 ˆ δφ √ ∂ φ c ( 2 4 symmetry was used to construct the order- ...m a 2 φ∂ − + 2 m 2 ∂, ∂ ¯ + q supergravity Lagrangian with cubic interaction vertices, ( = ∂ X = m − c ( ∂ 1 + φ 8(8) ( ∂ κ 2 m ∂ H (4 2 κ E 2 3 ǫ κ = 3 1 3 m dyn s , we prove that the new Lagrangian in ( × + ǫ d − δ B − h = φ δφ ], the φ = = 10 ′ ′ dyn s φ δ In [ Thus the To understand the action of SO(16) on the new superfield δφ to four dimensions while preserving the We now demonstrate how the “new” tranverse derivative, 6 Oxidation back to without cubic vertices and futher, both these versions have mensional reduction from metry transformations in We oxidize this expression to in a non-linear fashion. Finally, in appendix C, we prove tha In appendix We now note thatHamiltonian for using maximally the quadratic supersymmetric form theories expression one [ c invariant. such that where with JHEP01(2018)024 = ne = 4 = 4) , d (6.5) (6.4) since d sent in , d = 8 ). Since imensions operators, = 8 theory N = 8 2.2 m ¯ N d N , = 4) Yang-Mills m d symmetry, because , d or orm of ( 8(8) that the external states = 4 m , we oxidize the theory, E ¯ q ms of field components. l , s in the original paper on on from the = 8 supergravity in -invariant to this order we N st. This manifest enhanced m ward to write down. This is s which is both unitary and have been found to have the ws about q . In doing so, we need to take igher loops which the analysis N In some sense they have only 8(8) turbation series of ( clearly not the four-dimensional edefinition from the original one E φ = 4. Thus, we will end up with d . . m ∇ W m ≡ ǫ 2 ≡ i∂ – 9 – φ − do not contain any 1 ∂ = 4 with the same field content as in ( dyn s ∇ δ d ] that all of those amplitudes have the same divergence and 12 , which will introduce the conjugate “new” derivative ∇ ]).This perturbative simplicity is all the more remarkable m = 4 through the oxidation, we can in principle construct a 14 W d , -invariant. The answer is that our procedure, of going down o 13 in 8(8) = 4) supergravity. In order to argue that this symmetry is pre m E W , d = 8 is the dynamical supersymmetry variation on symmetry to this order. One could ask why we had to leave four d N m = 8 theory is unique, we have arrived at a form of 8(8) W E ] indicates. Note that in order to argue that the Hamiltonian is We have not discussed the supersymmetry generators in this f N 2 = 4 scattering amplitudes we must add up such amplitudes such span the full 128-dimensionalcomplete spinors. one-loop amplitudes We [ have seen though a This method of oxidation respects both the SO(16) and the ful d which can affect the invariance of the Hamiltonian in a maximal supergravity theory in states of ( must treat the states as 128-dimensional spinors. These are in [ pattern. It is a further assumption that this is true also to h the in the first place? We could have simply found a field redefiniti the generalized derivatives, where causal. There are also strong reasons to believe that the per 7 Conclusions Maximally supersymmetric gravity andsimplest Yang-Mills perturbation theories series amongthe theories bare of bone their structure kind. needed to build a perturbation serie supergravity. Since this formulation iswe obtained by do a not field expect r a supersymmetry price generators we to have be to straightfor pay in this formalism which is minimal in ter , allows us to rendersymmetry the is enhanced the symmetry difficult manife preserving the step enhanced to symmetry achieve. arriving at our Once goal. this is in place with to a form that is Once we obtain SO(16) invariant Hamiltonian with onlythe even complex order conjugate coupling of at least the Yang-Millsthrough the theory AdS-CFT non-perturbatively duality. even kno 4) supergravity is, in a certain sense, the square of that in ( theory (KLT-relations [ JHEP01(2018)024 ′ φ ′ ¯ φ ′ ¯ φ κ B + the A  ′ = ¯ φ ′ + φ ]. That analysis ∂ ) ′ × 19 − φ ]. Such symmetries /2014/000687). SM , 3 ∂ 21 = 4) supergravity is + 1 + 4  18 , ′ ∂ ∂ + return to. Not only do There have been strong tical constraints on the φ we keep terms which are ∂ , d  20 in these theories and we − + ey cannot be absorbed by 4 mmetries of the scattering κ 2 ∂ ′ + ∂ ould be invariant under the ¯ ( φ = 8 and maximal supersymmetry ng argument as was achieved on series ought be more finite 3 κ∂ space derivatives. However, we l symmetry-related inputs may  e symmetries as we have shown + 1 3 2 ′ N ) becomes ∂ φ 7(7) +  E 3.2 ) + ∂ ′ 4 ′ φ ¯ + φ + 3 ∂ κ∂ + 1 ′ 2 3 ∂ could be present [ φ +  11 ∂ 4 ) + ] and its deformations [ ( ′ E + κ ¯ φ – 10 – 17 3 1 + κ∂ ]). Counterterms could in principle be constructed ∂ and 3 4 ′ + φ ¯ ′ φ 15 ) 10 − φ + − ) E ∂ ′  ∂ ( φ ) + 4 κ ), the kinetic term in ( + ∂ − + 3 1 ]. We can only see two ways to finally settle the question ∂ ∂ ∂ ′ − , which are just complex conjugate of these terms, reproduce 5.4 ′ 16 + + 4 φ 2 φ ′ ∂ + + ′ ∂ ¯ ¯ φ φ ∂ ∂ ( ′ − ( ¯  ¯ φ φ ) 2 1 2 2 ∂ − ( − − ∂ + 4 only. = = ∂ + = 4 Yang-Mills theory [ ′ ∂ φ φ − ′ 4 N 2 φ + ′ ✷ ∂ ¯ ∂ φ ( ). ′ ¯ φ ¯ φ 3.2 − κ 3 2 − Are there even larger symmetries lurking in these theories? Our analysis does not shed light on whether the ( The other kind of terms 1 indications that the affine algebras The free order term gives back the kinetic term. Now, at order Under the field redefinition ( The work of SAacknowedges is support partially from supported a CSIR by NET a fellowship. DST-SERB grant (EMR A Verification of field redefinition amplitudes since thatamplitudes. would amount Those to symmetrieshope infinitely must to be many return more to kinema this deeply question ingrained in future publications. Acknowledgments help limit the possible diagrams that need checking. could possibly be realized byfind the it superfield difficult and to all its see super how such symmetries could directly be sy a field redefinition.residual Further, reparametrization, the local supersymmetry remainingin and counterterms gaug the sh case of(of pure finiteness): gravity do [ thein full the calculation case or of findcannot a be power carried counti over straightforwardly but new additiona in our formalism, butwe this need to is construct a counterterms, formidable we task must that also we prove hope that th to of the form vertex in ( suggest (for a related discussion, see [ perturbatively finite. We canthan only argue what that the the usual perturbati counterterm arguments based on JHEP01(2018)024 (B.1) (B.4) (B.3) ) ′ δφ ) φ ′ ( 4 2 φ ) ′ + + ✷ + φ ∂ ∂ ∂ ′ ′ + φ  φ ∂ 2 ) ′ − ∂ φ ′ , ∂ δφ ¯ − φ ( +  ∂ 4 ′ + ∂ c.c. . ) + ( + 1 φ ∂ ¯ + ∂ + ∂φ φ∂ ) ( ′ + ∂ 3 + ′ + φ ! + ∂ ∂ ′ + 1 ¯ φ ) (B.2) ′ ) ∂ ′ + ′ ′ ∂ 4 φ ∂ ¯ φ ∂φ ′ δφ 2 + 1 ∂φ  δφ 4 ( φ + ( ∂ + 4 + + 1 + ∂ − 4 ∂ κ , ∂ + ) ∂ ′ ∂ ) ∂ ′ + ′ ✷ 4 3 ( ¯ φ κ + ∂ c.c. . κ∂ 4 4 ′ δφ 4 3 ∂ δφ 2 3 ( ¯ + cubic ( ✷ φ ( + 1 + ′ ) + ∂ ∂ 2 ′ ∂ + − L + − ¯ ∂ + ) φ ∂ ) ′ δ ′ ′ ′ ) ∂ φ κ ∂φ − ¯ φ ′ ¯ φ φ φ ′ 4 3 2 + + ) ¯ 4 ′ φ 2 δφ 4 4 ′ φ + ∂ ( ✷ φ 4 + 1 + ′ 2 φ + + 1  ✷ ) ∂ 4 , we get ∂ ) + 1 ∂ ∂ ∂ + + ′ ) + − + κ ✷ ∂ ) ∂φ ∂ ∂  κ ′ φ φ ∂ – 11 – ′ ′ ∂ δφ + kinetic 2 + 2 ¯ φ 4 3 ( ′ φ ′ + φ φ + ∂ ∂ δ 4 + 2  L φ − ∂ + ( − 2 δ ∂ ∂ + φ ′ ✷ φ∂ ∂ ′ + + ) − ) − ∂ + + ¯ ′ ′ ′ + ¯ φ φ )( = ¯ ∂ ∂φ 2 φ φ φ φ ∂ 4 ′ δ = ′ − ¯ ( ∂ ( φ∂ + + + + φ + 1 ( ∂ L − 4 3 3 ∂ 2 ∂ ∂ ∂ ∂ ′ δ ′ ′ ′ + + + + 1 1 ∂ ✷  φ ′ ) φ φ φ ∂  ∂ ∂ ∂ 4 ′ 2 2 2 ∂φ φ ) ′ kinetic κ + ✷ −   ∂ ∂ ∂ ¯ + yields + φ δφ L 4 3 ( ∂ 4 4 δ + + 2 ∂ ′ ( ∂ ¯ δ ) ′ ( + ) 4 ∂ ∂ ∂ + + ′ ¯ L ¯ φ = φ κ ( ∂ ( ( ¯ ′ ′ ′ ′ + 1 φ δ 3 3 1 ¯ ¯ ¯ ( δ κ∂ κ∂ ∂ φ φ φ φ B + 1 ( 2 2 4 4 4 4 3 3 ( − 4 ∂

+ + + + + 1 1 1 1 terms are + +  + κ + 1  ∂ ∂ ∂ ∂ A 3 ∂ 4 κ κ κ κ κ κ = − 3 3 3 3 3 4 4 4 4 2 − − − = + = = + = + = = kinetic ′ ) and keeping terms up to order can be further simplified as follows. L B A cubic δ 5.7 ′ B L δ and B SO(16)-invariance of the newThe Lagrangian SO(16) variation of Hence, the order- where A and Using ( JHEP01(2018)024 fter (B.5) (C.1) (C.2) (C.3) ) ′ )] φ δφ . ′ 2 ( δ 2  ( ¯ + φ +   ∂ + ∂ ¯ ¯ ′ φ φ ) φ + + φ 2 φ∂ SO(16) transfor- ) ′ 1 ∂ 3 / δ φ ′ φ∂ φ∂ ′ ( + 1 ¯ φ 3 3 δ + ∂ 4 + + 1 1 8(8) c.c. , ∂ + 1 ∂ ∂  E + ∂ 4 −   + 4 4 ! φ + ′ ∂ + + ′ + ). We have only considered tion with a three-point cou-  ∂ ∂ φ ∂ ∂φ 2 s here by )   5.6 + ) takes the form φ + ] ′ 1 ∂ ′ 2 ∂ ) δ ) ′ ′ ′ 2 B.4 , δ SO(16) . Let us consider two coset trans- δ ′ 1 φ ( δφ SO(16) δ δφ ′ κδ ( [ ( + δ φ . 2 3 ∂ 2 κ κδ ∂ ∂ 3 2 + 3 2 ′ − ) ∂ − ¯ φ ′ ) − φ ) ¯ 4 φ φ SO(16) + φ ′ 4 + 1 = δ ′ 1 + φ ∂ + 1 δ φ∂ ( = ∂ φ∂ + ). We know that two such coset transformations + + – 12 – 2  φ + ∂ ′ in terms of ¯ ∂ ] φ ∂ ) ′ φ 2 ]( − 4.8 ′ ( + ′ 2 2 φ ′ φ , δ ′ 2 ′ + ′ 1 , δ δ δ φ∂ Y ∂ ∂φ ′ δ ( 1 ′ 3 [ δ + + symmetry. The action of the 128 [ φ + + 1 SO(16) ∂ 2 ∂ κ ′ ∂ ∂ X 1 3 ) κδ ) ) ) +  8(8) ′ φ 3 1 ∂φ φ φ + 4 − φ E + + and simple manipulations, ( + + is given in ( δφ , + φ + − ) rendering the new Lagrangian, with a cubic vertex, SO(16)- φ ∂ ( ′ ¯ ′ ] ∂ ) φ + ∂ φ∂ 4 ′ φ ′ φ φ φ∂ φ∂ 2 ) ] ∂  ¯ + + 1 φ ), since the others are contained in the complex conjugate. A + ′ B.3 2 φ ] , δ cancel against each other, as in eq. ( δ ∂ ∂ ′ 2 ′ ∂ 2 ′ 1 on ( ( , δ δ SO(16) 0 δ ]( 4 ′ ′

1 1 δ ′ 2 , δ ¯ ′ κ 2 φφφ SO(16) SO(16) δ δ ′ δ 1 κ + 1 [ ( δ δ δ 3 ∂ 4 = = [ , δ [ + + ′ 1 ′ simplify to κ SO(16) ∂ = = − − δ ∂ [ [ ( φ 2 3 and ′ ] Y κ κ κ κδ ′ 2 = ′ φ 1 ) we find that for the new field − ] 1 3 3 3 3 2 1 2 δ ′ 2 , δ ′ = 1 ), we can readily express − − − − 5.7 , δ and δ invariance [ ′ 1 Y = = = = δ 5.7 ′ kinetic 8 [ X L X E δ Using ( where partially integrations of and So, from ( mations on the superfield C should close on SO(16), (we denote the coset transformation invariant. terms of the form ( We show here, how thepling non-linearly can realised be SO(16) extended for to the an ac which cancels against ( formations, The terms of order thus proving that the transformations close. JHEP01(2018)024 ]. ]. , ] SPIRE SPIRE ommons IN IN [ ][ (2008) 113 = 4 , N 07 ]. (2013) 118 (2008) 034 (1979) 141 Supergravity as 03 , JHEP SPIRE 06 , IN = 8 ]. arXiv:1601.02836 (2011) 074 [ ][ Amplitudes and Ultraviolet ]. JHEP N B 159 arXiv:1103.1848 , 08 Supergravity Hamiltonian as a [ JHEP redited. , ]. ]. SPIRE iban, ]. SPIRE = 8 IN [ IN [ JHEP N SPIRE SPIRE (2016) 051 , IN IN SPIRE ]. Nucl. Phys. (2011) 561 03 The IN Yang-Mills Theory on the Light Cone ][ ][ , . hep-th/0607019 ][ [ Yang-Mills and 59 Cubic Interaction Terms for Arbitrarily = 4 (1982) 474 (1983) 41 SPIRE JHEP Counterterms in Gravity in the Light-Front The Ultraviolet Finiteness of the N = 4 IN on the Light Cone in Light Cone Superspace [ , A Relation Between Tree Amplitudes of Closed and N – 13 – ]. ]. Exceptional versus superPoincar´ealgebra as the (1983) 323 B 198 7(7) 8(8) B 227 Supergravity Eleven-dimensional supergravity in light-cone Eleven-dimensional supergravity in light-cone E E (2006) 195 (1986) 1 hep-th/0501079 hep-th/0501079 SPIRE SPIRE [ [ Fortsch. Phys. IN IN B 123 , SO(8) [ [ arXiv:0706.1778 [ Three-Dimensional Supergravity Theories ), which permits any use, distribution and reproduction in B 753 ]. ]. ]. ]. Counterterms vs. Dualities KLT relations from the Einstein-Hilbert Lagrangian Nucl. Phys. The B 269 Conformal-like Symmetry in Gravity Nucl. Phys. , , SPIRE SPIRE SPIRE SPIRE (2005) 003 (2005) 003 = 2 IN IN IN IN Phys. Lett. (1983) 145 (1983) 401 Supergravity D (2007) 128 05 05 ][ ][ ][ ][ , CC-BY 4.0 Nucl. Phys. , = 8 This article is distributed under the terms of the Creative C Nucl. Phys. , N B 228 B 212 JHEP JHEP B 652 , , ]. SPIRE arXiv:1212.2776 arXiv:1105.1273 arXiv:0801.2993 arXiv:0804.4300 IN Limits of Theories Yang-Mills Theory superspace Formulation and a [ Open Strings Phys. Lett. [ superspace [ [ Nucl. Phys. Extended Supermultiplets [ Nucl. Phys. defining symmetry of maximal supergravity Behavior of Quadratic Form [1] E. Cremmer and B. Julia, [4] S. Ananth, L. Brink and P. Ramond, [9] L. Brink, S.-S. Kim and P. Ramond, [6] L. Brink, O. Lindgren and B.E.W. Nilsson, [7] A.K.H. Bengtsson, I. Bengtsson and L. Brink, [8] S. Ananth, L. Brink and P. Ramond, [5] N. Marcus and J.H. Schwarz, [2] Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson[3] and R. S. Ro Ananth, L. Brink and S. Majumdar, [16] A.K.H. Bengtsson, L. Brink and S.-S. Kim, [17] L. Brink, O. Lindgren and B.E.W. Nilsson, [14] S. Ananth and S. Theisen, [15] G. Bossard and H. Nicolai, [11] S. Ananth, L. Brink, R. Heise and H.G.[12] Svendsen, M.B. Green, J.H. Schwarz and L. Brink, [13] H. Kawai, D.C. Lewellen and S.H.H. Tye, [10] L. Brink, S.-S. Kim and P. Ramond, any medium, provided the original author(s) and source are c References Open Access. Attribution License ( JHEP01(2018)024 , ] ] hep-th/0702020 [ hep-th/0104081 [ ]. ]. (2007) 227 SPIRE SPIRE (2001) 4443 IN IN ][ B 783 18 ][ and a ’small tension expansion’ of M-theory 10 E – 14 – Nucl. Phys. Proof of all-order finiteness for planar beta-deformed Proof of ultra-violet finiteness for a planar , hep-th/0207267 [ hep-th/0609149 [ Class. Quant. Grav. , (2007) 046 (2002) 221601 01 89 and M-theory 11 JHEP E , ]. ]. SPIRE SPIRE IN IN [ Yang-Mills non-supersymmetric Yang-Mills theory [ Phys. Rev. Lett. [20] T. Damour, M. Henneaux and H. Nicolai, [21] P.C. West, [19] S. Ananth, S. Kovacs and H. Shimada, [18] S. Ananth, S. Kovacs and H. Shimada,