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AEI–2015–005, DAMTP–2015–5 A solution to the non-linear equations of D=10 super Yang–Mills theory Carlos R. Mafra1,∗ and Oliver Schlotterer2,† 1DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK 2Max–Planck–Institut fu¨r Gravitationsphysik, Albert–Einstein–Institut, 14476 Potsdam, Germany (Dated: September22, 2015) In this letter, we present a formal solution to the non-linear field equations of ten-dimensional super Yang–Mills theory. It is assembled from products of linearized superfields which have been introduced as multiparticle superfields in the context of superstring perturbation theory. Their explicit form follows recursively from the conformal field theory description of the gluon multiplet in thepurespinor superstring. Furthermore, superfieldsof highermass dimensions aredefined and theirequations of motion spelled out. 5 1 0 INTRODUCTION group. The derivatives are taken with respect to ten- 2 dimensionalsuperspacecoordinates(xm,θα)withvector p SuperYang–Mills(SYM)theoryintendimensionscan andspinorindicesm,n=0,...,9andα,β =1,...,16of e beregardedasoneofthesimplestSYMtheories,itsspec- the Lorentz group. The fermionic covariant derivatives S trum contains just the gluon and gluino, related by six- 9 teen supercharges. However,it is well-knownthat its di- Dα ≡∂α+ 21(γmθ)α∂m, {Dα,Dβ}=γαmβ∂m (2) 1 mensional reduction gives rise to various maximally su- involve the 16×16 Pauli matrices γm = γm subject to persymmetric Yang–Mills theories in lower dimensions, αβ βα h] including the celebrated N = 4 theory in D = 4 [1]. theCliffordalgebraγα(mβγn)βγ =2ηmnδαγ,andtheconven- -t Therefore a better understanding of this theory propa- tion for (anti)symmetrizing indices does not include 21. p gates a variety of applications to any dimension D ≤10. The connections in (1) give rise to field-strengths e Inarecentlineofresearch[2,3],scatteringamplitudes h F ≡{∇ ,∇ }−γm∇ , F ≡−[∇ ,∇ ]. (3) [ of ten-dimensional SYM have been determined and sim- αβ α β αβ m mn m n 2 plified using so-called multiparticle superfields [4]. They One canshowthatthe constraintequationFαβ =0 puts represent entire tree-level subdiagrams and build up in v thesuperfieldson-shell,andBianchiidentitiesleadtothe 2 the conformal field theory (CFT) on the worldsheet of non-linear equations of motion [8], 6 the pure spinor superstring [5] via operator product ex- 5 pansions (OPEs). Multiparticle superfields satisfy the ∇ ,∇ =γm∇ 5 α β αβ m 0 ltienremarsi,zie.ed. ifinevlderseeqouffat-siohnesllwpriothpatghaetoards.dIitniotnhisoflectotenrtawcet (cid:8)∇α,∇m(cid:9)=−(γmW)α 1. demonstrate that these off-shell modifications can be re- (cid:2)∇α,Wβ(cid:3)= 14(γmn)αβFmn 50 summed to capture the non-linearitiesin the SYM equa- (cid:8)∇α,Fmn(cid:9)= ∇[m,(γn]W)α . (4) 1 tions of motion. The generating series of multiparticle (cid:2) (cid:3) (cid:2) (cid:3) : superfieldsasseenin(18)isshowntosolvethenon-linear In the subsequent, we will construct an explicit solution v field equations spelled out in (4). for the superfields A , A , Wα and Fmn in (4). The i α m X We alsodefine superfields of arbitrarymass dimension main result is furnished by the generating series in (18) r and reduce their non-linear expressions to the linearized whoseconstituentswillbeintroducedinthenextsection. a superfields of lower mass dimensions. This framework simplifies the expressions of kinematic factors in higher- loopscatteringamplitudes, including theD6R4 operator LINEARIZED MULTIPARTICLE SUPERFIELDS in the superstring three-loop amplitude [6]. In perturbation theory, it is conventional to study solutions A ,A ,... of the linearized equations of motion α m REVIEW OF TEN-DIMENSIONAL SYM D ,A =γmA (α β) αβ m (cid:8) (cid:9) The equations of motion of ten-dimensionalSYM the- D ,A =k A +(γ W) α m m α m α ory can be described covariantly in superspace by defin- (cid:2)D ,Wβ(cid:3)= 1(γmn) βF ing supercovariant derivatives [7, 8] α 4 α mn (cid:8) (cid:9) D ,Fmn =k[m(γn]W) . (5) α α ∇α ≡Dα−Aα(x,θ), ∇m ≡∂m−Am(x,θ). (1) (cid:2) (cid:3) Their dependence on the bosonic coordinates x is de- The connections A and A take values in the Lie al- scribed by plane waves ek·x with on-shell momentum α m gebra associated with the non-abelian Yang–Mills gauge k2 =0. In a gaugewhere θαA =0, the θ dependence is α 2 knownintermsoffermionicpowerseriesexpansionsfrom pole structure of tree-level subdiagrams in SYM theory [9,10]whose coefficients containgluonpolarizationsand obtained from the point-particle limit. gluino wave functions. The CFT-inspiredtwo-particleprescription(6) canbe As an efficient tool to determine and compactly rep- promoted to a recursion leading to superfields of arbi- resent scattering amplitudes in SYM and string theory, trary multiplicity, see (3.54), (3.56) and (3.59) of [4]. multiparticle versions of the linearized superfields have Their equations of motion are observed to generalize been constructed in [4]. They satisfy systematic modi- along the lines of fications of the linearized equations of motion (5), and their significance for BRST invariance was pointed out D(α,A1β2)3 =γαmβA1m23+(k12·k3) A1α2A3β −(12↔3) in [11]. For example, their two-particle version (cid:8) (cid:9)+(k1·k2) A1A23+A13(cid:2)A2 −(1↔2) (10)(cid:3) α β α β (cid:2) (cid:3) A1α2 ≡−12 A1α(k1·A2)+A1m(γmW2)α−(1↔2) forsuitabledefinitionsofA1α23andA1m23,see(3.17),(3.19) A12 ≡ 1 A(cid:2)pF2 −A1 (k1·A2)+(W1γ W2)−((cid:3)1↔2) and (3.29) of [4]. m 2 1 pm m m Wα ≡ 1(cid:2)(γmnW )αF1 +Wα(k2·A1)−(1↔2) (6)(cid:3) Multiparticle superfields can be arranged to satisfy 12 4 2 mn 2 kinematic analoguesof the Lie algebraicJacobirelations F12 ≡F2 (k2·A1)+ 1F2 pF1 among structure constants, e.g. A123+A231+A312 =0. mn mn 2 [m n]p α α α +k1 (W1γ W2)−(1↔2), They therefore manifest the BCJ duality [12] between [m n] colorandkinematicdegreesoffreedominscatteringam- canbecheckedvia(5)tosatisfythefollowingtwo-particle plitudes, see [13] for a realization at tree-level. Together equations of motion: with the momenta k12...j ≡ k1 +k2 +...+kj in their equationsofmotion,thissuggeststoassociatethemwith {D ,A12}=γmA12+(k1·k2)(A1A2 +A1A2) (7) tree-levelsubdiagramsshowninthesubsequentfigure[4]: (α β) αβ m α β β α {[DD[Dαα,,,WFA11βm1222}]]===kγ41α1m(2γβ(Wmγn1βW)2α+β1k2F)1mm122An−1α+2k+(1k2((1kγ1·k·W2k)2(1)A2()A1αW1αA2βm2−−AA2α2αWA1m1β)) 2 k132 k1243 ...p k12....p.. ↔ WAA1α1m1α222333.........ppp α mn m n α n m α 1 F123...p mn +(k1·k2) A1αFm2n+A1[n(γm]W2)α−(1↔2) .  The cubic-graph organization of superfields already ac- (cid:0) (cid:1) Themodificationsascomparedtothesingle-particlecase counts for the quartic vertex in the bosonic Feynman (5) involve the overall momentum k ≡ k +k whose rules of SYM. This ties in with the string theory origin 12 1 2 propagator is generically off-shell, k2 =2(k ·k )6=0. of n-particle tree-level amplitudes where each contribu- 12 1 2 The construction of the two-particle superfields in (6) tion stems from n−3 OPEs. is guided by string theory methods. In the pure spinor Berends–Giele currents: As a convenient basis of formalism [5], the insertion of a gluon multiplet state on multiparticlefieldsKB ∈{ABα,AmB,WBα,FBmn}withmul- the boundary of an open string worldsheet is described tiparticlelabelB =12...p,wedefineBerends–Gielecur- by the integrated vertex operator rents KB∈{ABα,AmB,WBα,FBmn}, e.g. K1 ≡K1 and [4] Ui ≡∂θαAiα+ΠmAim+dαWiα+ 12NmnFmin . (8) K12 ≡ Ks12 , K123 ≡ sKs123 + sKs321 (11) 12 12 123 23 123 Worldsheet fields [∂θα,Πm,d ,Nmn] with conformal α withgeneralizedMandelstaminvariantss ≡ 1k2 . weightone andwell-knownOPEsarecombinedwithlin- 12...p 2 12...p Berends–Giele currents K are defined to encompass all earized superfields associated with particle label i. The B treesubdiagramscompatiblewiththeorderingoftheex- multiplicity-two superfields in (6) are obtained from the ternal legs in B. The propagators s−1 absorb the ap- coefficients of the conformal fields in the OPE [4] i...j pearanceofexplicitmomentainthe contacttermsofthe equations of motion such as (7) and (10). U12 ≡− (z −z )α′k1·k2U1(z )U2(z ) (9) I 1 2 1 2 Asillustratedinthefollowingfigure,thethree-particle =∂θαA12+ΠmA12+d Wα + 1NmnF12 , current in (11) is assembled from the s- and t-channels α m α 12 2 mn of a color-ordered four-point amplitude with an off-shell where α′ denotes the inverse string tension, and total leg (represented by ...): derivativesintheinsertionpointsz ,z ontheworldsheet 2 3 3 1 2 s s have been discarded in the second line. The contour in- K = 12 ··· + 123... 123 s s tegral in (9) isolates the singular behaviour of the Ui 123 23 w.r.t. (z −z ) which translates into propagatorsk−2 of 1 2 1 1 2 12 the gauge theory amplitude after performing the α′ →0 IncontrasttothebosonicBerends–Gielecurrentsin[14], limit. In other words, OPEs in string theory govern the the currents K manifest maximal supersymmetry, and B 3 their construction does not include any quartic vertices. 0=K =K +K +K (16) 1 23 123 213 231 (cid:1) Aclosedformulaatarbitrarymultiplicity [4,15]involves 0=K −K =K −K , 12 3 1 32 123 321 the inverse of the momentum kernel1 S[·|·] [16], (cid:1) (cid:1) 1 and symmetries (15) at higher multiplicity leave (p−1)! K ≡ S−1[σ|ρ] K , (12) 1σ(23...p) 1 1ρ(23...p) independent permutations of K . Any permutation X 12...p ρ∈Sp−1 can be expanded in a basis of K with σ ∈ S 1σ(23...p) p−1 with permutation σ ∈Sp−1 of legs 2,3,...,p. through the Berends–Giele symmetry The combination of color-ordered trees as in (11) and (12)simplifies theirmultiparticle equationsofmotion[4] K =(−1)|B|K , (17) B1A 1(A Bt) (cid:1) D(α,ABβ) =γαmβABm+ AXαAYβ −AYαAXβ (13) where |B| = p and Bt = bp...b2b1 for a multi particle (cid:8) (cid:9) XXY=B(cid:0) (cid:1) label B = b b ...b . Since the Berends–Giele current 1 2 p Dα,ABm =kmBABα +(γmWB)α+ AXαAYm−AYαAXm K12p is composed from the cubic diagrams in a partial (cid:2) (cid:3) XXY=B(cid:0) (cid:1) amplitude with an additional off-shell leg, (17) can be D ,Wβ = 1(γmn) βFB + AXWβ −AYWβ understood as a Kleiss–Kuijf relation among the latter (cid:8) α B(cid:9) 4 α mn XXY=B(cid:0) α Y α X(cid:1) [24]. D ,Fmn =k[m(γn]W ) + AXFmn−AYFmn α B B B α α Y α X (cid:2) (cid:3) XXY=B(cid:0) (cid:1) GENERATING SERIES OF SYM SUPERFIELDS + A[n(γm]W ) −A[n(γm]W ) . X Y α Y X α XXY=B(cid:0) (cid:1) In order to connect multiparticle fields and Berends– Momenta kB ≡ k1 + k2 + ...kp are associated with Giele currentswith the non-linearfieldequations (4), we multiparticle labels B = 12...p, and instructs define generating series K∈{A ,Am,Wα,Fmn} XY=B α to sum over all their deconcatenationPs into non-empty X = 12...j and Y = j +1...p with 1 ≤ j ≤ p−1. K≡ K ti+ K titj + K titjtk+... (18) i ij ijk E.g. the three-particle equation of motion of A123 reads X X X α i i,j i,j,k D(α,A1β2)3 =γαmβA1m23 (14) = Kiti+ 21 Kij[ti,tj]+ 31 Kijk[[ti,tj],tk]+··· (cid:8) (cid:9) X X X +A1A23+A12A3 −A23A1 −A3A12 , i i,j i,j,k α β α β α β α β and a comparisonwith (10) highlights the advantagesof where ti denote generatorsin the Lie algebraofthe non- the diagramexpansions in (11). Superfields up to multi- abeliangaugegroup. Hence,thegeneratingseriesin(18) plicity five satisfying (13) were explicitly constructed in adjoin color degrees of freedom to the polarization and [4] and there are no indications of a breakdown of (13) momentumdependenceinthelinearizedmultiparticlesu- at higher multiplicity. perfieldsK . Thesecondlinefollowsfromthesymmetry B The symmetry properties of the K can be inferred (15), which guarantees that K is a Lie element [17]. B fromtheircubic-graphexpansionandsummarizedas[23] As a key virtue of the series K ∈ {A ,Am,Wα,Fmn} α in (18),they allowto rewrite the D actiononBerends– K =0, ∀A,B 6=∅ , (15) α A(cid:1)B GielecurrentsKB ∈{ABα,AmB,WBα,FBmn}in(13)asnon- where denotes the shuffle product2 [17]. For example, linear equations of motion, (cid:1) 0=K =K +K 1 2 12 21 D ,A =γmA +{A ,A } (cid:1) (α β) αβ m α β (cid:8) D ,A (cid:9)= ∂ ,A +(γ W) +[A ,A ] α m m α m α α m (cid:2)D ,Wβ(cid:3)=(cid:2)1(γmn)(cid:3)βF +{A ,Wβ} 1 Themomentumkernelisdefinedby[16] (cid:8) α (cid:9) 4 α mn α D ,Fmn = ∂[m,(γn]W) +[A ,Fmn] p j−1 α α α (cid:2) (cid:3) (cid:2) (cid:3) S[σ(2,3,...,p)|ρ(2,3,...,p)]1= Y(s1,jσ +Xθ(jσ,kσ)sjσ,kσ) −[A[m,(γn]W)α] , (19) j=2 k=2 anddependsonreferenceleg1andtwopermutationsσ,ρ∈Sp−1 where [∂m,K] translates into components kBmKB. of additional p−1 legs 2,3,...,p. The symbols θ(jσ,kσ) keep Remarkably,theyareequivalenttothenon-linearSYM trackoflabelswhichswaptheirrelativepositionsinthetwoper- field equations (4) if the connection in (1) is defined mutationsσ andρ,i.e. θ(jσ,kσ)=1(=0)iftheorderingofthe legsjσ,kσisthesame(opposite)intheorderedsetsσ(2,3,...,p) through the representatives Aα and Am of the generat- andρ(2,3,...,p). Theinversein(12)istakenw.r.t.matrixmul- ing series in (18). In other words, the resummation of tiplicationwhichtreatsσ andρasrow-andcolumnindices. linearized multiparticle superfields {AB,Am,Wα,Fmn} 2 mThuetasthiounfflseσporfodAu∪ctBinwKhiAc(cid:1)hBpreissedrevfientehdetoordseurmofatllheKiσndfoivridpuear-l through the generating series (18) of BαerenBds–GBieleBcur- elementsofbothsetsAandB. rents (12) solves the non-linear SYM equations (4). 4 Given that the multiparticle superfields satisfy [4] where the vertical bar separates the antisymmetric pair of indices present in the recursion start Fpq. Their com- Fmn =k[mAn]− (AmAn −AmAn) (20) ponent fields are defined by B B B X Y Y X X XY=B kmB(γmWB)α = AXm(γmWY)α−AYm(γmWX)α Wm1...mkα ≡ XtBWBm1...mkα, (27) XXY=B(cid:2) (cid:3) B6=∅ kmBFBmn = 2(WXγnWY)+AXmFYmn−AYmFXmn , Fm1...mk|pq ≡ tBFBm1...mk|pq, XXY=B(cid:2) (cid:3) BX6=∅ the above definitions are compatible with (3) and with tB ≡ t1t2...tp for B = 12...p. They inherit the Berends–Giele symmetries (15) and can be identified as ∇ ,(γmW) =0, ∇ ,Fmn =γn Wα,Wβ , m α m αβ (cid:2) (cid:3) (cid:2) (cid:3) (cid:8) (cid:9)(21) Wm1...mkα =km1Wm2...mkα (28) B B B whose lowest components in θ encode the Dirac and + Wm2...mkαAm1 −Wm2...mkαAm1 , Yang-Mills equations for the gluino and gluon field. X Y Y X A linearized gauge transformation in particle one, XXY=B(cid:0) (cid:1) Fm1...mk|pq =km1Fm2...mk|pq B B B δ1A1α =DαΩ1, δ1A1m =km1 Ω1, (22) + Fm2...mk|pqAm1 −Fm2...mk|pqAm1 . X Y Y X can be described by a scalar superfield Ω , it shifts the XXY=B(cid:0) (cid:1) 1 gluon polarization by its momentum. The gluino com- Note from (28) that the non-linearities in the defini- ponent and the linearized field strengths are invariant tion of higher mass superfields do not contribute in the under (22), δ1W1α =δ1F1mn =0, whereas Berends–Giele single-particle context with Wm1...mkα = km1Wm2...mkα i i i currents of multiparticle superfields with B = 12...p whereas the simplest two-particle correction reads transform as follows [18]: Wmα =kmWα +WαAm−WαAm . (29) 12 12 12 1 2 2 1 δ AB = D ,Ω + Ω AY, δ Wα = Ω Wα 1 α α B X α 1 B X Y (cid:2) (cid:3) XXY=B XXY=B Equations of motion at higher mass dimension: δ AB = ∂ ,Ω + Ω AY, δ Fmn = Ω Fmn. 1 m m B X m 1 B X Y Theequationsofmotionforthesuperfieldsofhighermass (cid:2) (cid:3) XXY=B XXY=B dimension (26) follow from ∇ ,∇ = −(γ W) and α m m α (23) ∇ ,∇ =−F togetherw(cid:2)ithJaco(cid:3)biidentitiesamong m n mn (cid:2)iterated b(cid:3)rackets. The simplest examples are given by The multiparticle gaugescalarsΩ areexemplifiedin 12...p appendix B of [18] and gathered in the generating series ∇ ,Wmβ = 1(γ ) βFm|pq− (Wγm) ,Wβ , α 4 pq α α (cid:8) (cid:9) (cid:8) (cid:9) L1 ≡Ω1t1+ Ω1i[t1,ti]+ Ω1jk[[t1,tj],tk]+... (24) ∇α,Fm|pq =(Wm[pγq])α− (Wγm)α,Fpq , (30) Xi Xj,k (cid:2) (cid:3) (cid:2) (cid:3) which translate into component equations of motion This allows to cast (23) into the standard form of non- linear gauge transformations, D Wmβ = 1(γ ) βFm|pq+ (AXWmβ −AYWmβ) α B 4 pq α B α Y α X X XY=B δ1Aα = ∇α,L1 , δ1Wα = L1,Wα (25) − (W γm) Wβ −(W γm) Wβ , δ1Am =(cid:2)∇m,L1(cid:3), δ1Fmn =(cid:2)L1,Fmn(cid:3) , XXY=B(cid:2) X α Y Y α X(cid:3) (cid:2) (cid:3) (cid:2) (cid:3) D Fm|pq =(Wm[pγq]) + (AXFm|pq−AYFm|pq) such that traces w.r.t. generators ti furnish a suitable α B B α α Y α X X XY=B starting point to construct gauge invariants. − (W γm) Fpq−(W γm) Fpq . (31) X α Y Y α X XXY=B(cid:2) (cid:3) HIGHER MASS DIMENSION SUPERFIELDS In general, one can prove by induction that TheintroductionoftheLieelementsKandtheirasso- ∇ ,WNβ = 1(γ ) βFN|pq− (Wγ)M,W(N\M)β ciated supercovariantderivatives allow the recursivedef- α 4 pq α α (cid:8) (cid:9) MX∈P(N(cid:8)) (cid:9) inition of superfields with higher mass dimensions, M6=∅ ∇ ,FN|pq =(WN[pγq]) − (Wγ)M,F(N\M)pq . Wm1...mkα ≡ ∇m1,Wm2...mkα , (26) (cid:2) α (cid:3) α MX∈P(N)(cid:2) α (cid:3) Fm1...mk|pq ≡(cid:2)∇m1,Fm2...mk|pq(cid:3), M6=∅ (32) (cid:2) (cid:3) 5 The vector indices have been gathered to a multi-index and generalizations to superamplitudes with a single in- N ≡ n n ...n . Its power set P(N) consists of the 2k sertion of supersymmetrized operators F4 or D2F4 will 1 2 k ordered subsets, and (Wγ)N ≡(Wn1...nk−1γnk). be given elsewhere [22]. Thehigher-mass-dimensionsuperfieldsobeyfurtherre- The multiparticle superfields of higher mass dimen- lations which can be derived from Jacobi identities of sionscanbeusedtoobtainsimplerexpressionsforhigher- nested (anti)commutators. For example, (3) determines loop kinematic factors of superstring amplitudes. For their antisymmetrized components example, the complicated three-loop kinematic factors generatingthematrixelementofthe(supersymmetrized) W[n1n2]n3...nkβ = Wn3...nkβ,Fn1n2 (33) operator D6R4 [6] (cid:2) (cid:3) F[n1n2]n3...nk|pq = Fn3...nk|pq,Fn1n2 . Moreover, the definitions (2(cid:2)6) via iterated co(cid:3)mmutators MD6R4 = |T1s2,3,4|2 +|T1m234|2+(1,2|1,2,3,4) (38) 12 imply that can be equivalently represented by F[m|np] =0, F[mn]|pq+F[pq]|mn =0, (34) andthegauge-variationsδ1∇m =[L1,∇m]and(25)yield T12,3,4 ≡h(λγmW1n2)(λγnW[p3)(λγpW4m])i δ Wα = L ,Wα , δ FN|pq = L ,FN|pq . (35) T1m234 ≡hAm(1T2),3,4+(λγmW(1)L2),3,4i (39) 1 N 1 N 1 1 (cid:2) (cid:3) (cid:2) (cid:3) L ≡ 1(λγnWq)(λγqWp)Fnpi . Manifold generalizations of (21), (33) and (34) can be 2,3,4 3 [2 3 4] generated using these same manipulations. In (38), the notation (A ,...,A | A ,...,A ) instructs 1 p 1 n to sum over all possible ways to choose p elements A ,A ,...,A out of the set {A ,...,A }, for a total of OUTLOOK AND APPLICATIONS 1 2 p 1 n n terms. Thetensorproductsofleft- andright-moving p (cid:0)SY(cid:1)M superfields in |T |2 = T T˜ are under- Therepresentationofthenon-linearsuperfieldsoften- 12,3,4 12,3,4 12,3,4 stood to yield superfields of type IIB or type IIA super- dimensionalSYMtheorydescribedinthisletterwasmo- gravity. Accordingly,the componentpolarizationsof the tivated by the computation of scattering amplitudes in supergravity multiplet arise from the tensor product of the pure spinorformalism. Accordingly,they giverise to gluon polarizations and gluino wavefunctions within the generating functions for amplitudes. For example, color- SYM superfields. dressed SYM amplitudes at tree-level M(1,2,...,n) in- The low-energy limit of the three-loop closed string volving particles 1,2,...,n are generated by amplitude given in (38) is proportional to the (α′)6 cor- ∞ 1TrhVVVi= (n−2) M(i ,i ,...,i ) . (36) rection of the corresponding tree-level amplitude which 3 1 2 n in turn defines the aforementioned D6R4 operator. X X n=3 i1<i2<...<in It would be interesting to explore the dimensional re- As firstly pointed out in the appendix of [19], the gen- duction [1] of the above setup and its generalization to erating series V ≡ λαAα involving the pure spinor λα SYM theories with less supersymmetry or to construct satisfies the field equations QV = VV of the action formal solutions to supergravity field equations along Tr d10xh1VQV−1VVVi[20]withBRSToperatorQ≡ similar lines. 2 3 λαDR . The zero mode prescription of schematic form α hλ3θ5i=1 extracting the gluon- and gluino components Acknowledgements: We thank Eduardo Casali, Evgeny Skvortsov, Massimo Taronna and Stefan Theisen for is explained in [5] and automated in [21]. With n=3 or valuablediscussionsaswellasNathanBerkovitsforhelp- n = 4 external states, for instance, assembling the com- ponentsVB ≡λαAB ofVVVin(36)withtheappropriate ful comments on the draft. This research has received α funding from the European Research Council under the number of labels yields EuropeanCommunity’s SeventhFrameworkProgramme M(1,2,3)=Tr(t1[t2,t3])hV1V2V3i (37) (FP7/2007-2013)/ERCgrant agreement no. [247252]. V12V3V4 V23V4V1 2M(1,2,3,4)=Tr(t1t2t3t4) + D s12 s23 V34V1V2 V41V2V3 + + +perm(2,3,4) . s34 s41 E ∗ Electronic address: [email protected] † Electronic address: [email protected] The pure spinor representation of ten-dimensional3 n- [1] L.Brink,J.H.SchwarzandJ.Scherk,Nucl.Phys.B121, pointSYMamplitudesisdescribedin[2]. 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