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JHEP06(2014)088 Springer April 2, 2014 May 26, 2014 June 16, 2014 : : : n-Simons Theo- f the tree-level S- Received Accepted Published 10.1007/JHEP06(2014)088 neralization of this theory itary gauge group. We also doi: ree-level S-matrix — which has h can be dimensionally-reduced, raction terms. Our construction Published for SISSA by [email protected] = 6 super-Chern-Simons theory — is due to conformal super- , N . 3 1401.6242 The Authors. c Scattering Amplitudes, Supersymmetric gauge theory, Cher

We construct an open theory whose single-trace part o , [email protected] University Park, PA 16802, U.S.A. E-mail: Department of , The Pennsylvania State University, Open Access Article funded by SCOAP Abstract: Oluf Tang Engelund and Radu Roiban A twistor string for the ABJ(M) theory demonstrate that the multi-trace partno of counterpart the in string the theory pure t matrix reproduces the S-matrix of the ABJ(M) theory with a un ArXiv ePrint: to product gauge groups with more than two factors. Keywords: ries interactions and identifysuggests certain that Lagrangian there exists inte ain higher dimensional a theory certain whic sense, to the ABJ(M) theory. It also suggests a ge JHEP06(2014)088 8 1 3 6 10 17 18 11 16 20 hile ] of the 9 e standard The discon- 1 = 4 super-Yang-Mills N ]. ], while making their factor- 15 4 presentation of amplitudes of evel y on shell), relations between ories have many different pre- iance (at tree-level and at the pertwistor space. 1 ] suggest that there may exist a , oles on which they factorize but struction of amplitudes based on 4 | mber of dimensions, has recently been ons [ 13 2 , 2 ]. The Grassmaniann presentation [ ]. It also exposes new properties, such as the 8 5 – 1 – ] makes manifest its super-conformal symmetry, ] while obscuring locality. Last but not least the 3 1) global symmetry [ , 12 – and its truncation to ABJ(M) states 10 1 , 4 | 2 , 2 ]. 7 , 6 ] and its open string formulation [ ] for a review) and each of them exposes different properties w 2 1 = 4 conformal whose action, differently from th N ] in the connected prescription [ 2 ], does not have a manifest SU(1 14 In contrast, Witten’s twistor string formulation of tree-l A different localization of (s)YM amplitudes, valid in any nu 5.1 The four-point5.2 amplitudes CGS interactions for any number of external legs 3.1 Vertex operators3.2 with reduced symmetry The general structure of scattering amplitudes 1 theory [ twistor string theory [ formulation of localization on curves ofnected a formulation certain implies degree the in existence themaximally-helicity-violating of building a relevant recursive su blocks con [ same amplitudes manifests theirlevel dual of superconformal the integrand invar at loop level) [ obscures most symmetries (especiallycolor-ordered those amplitudes, that etc. emerge onl the U(1) decoupling identities andization the Kleiss-Kuijff properties relati quite difficult to identify [ potentially obscuring others. For example,gauge a theories Feynman graph manifests locality and (at tree-level) the p proposed and discussed in [ Quite generally, scattering amplitudessentations of (see quantum field e.g. the [ 1 Introduction 5 ABJ(M) coupling to conformal supergravity 6 Outlook 3 A string theory on CP 4 ABJ(M) amplitudes from the truncated CP Contents 1 Introduction 2 An embedding of the 3d one [ JHEP06(2014)088 4 n 1) ]. nal , = 4 4 23 | ] along 2 N , gravity 25 ] and to cluded in eory have 30 – = 4 sYM tree- ]. 28 N ] and invariance 25 18 , 17 and four-dimensional groups [ ] was recently shown to be 23 ematics determining the ] and therefore does not ng theory which has the etry group, the former has ional reduction. 27 eted as describing a higher- ghost fields. Alternatively, [ principle be imposed at the rresponding Lagrangian. It ] exhibiting holomorphic lo- es, they combine into repre- r from scattering amplitudes ll construct has SU(3 actions. which in turn suggests that a 23 a string theory would also con- at space theory. Consequently, r the Yangian of the supercon- to identify tree-level S-matrices tering amplitudes of this theory ated expressions with those fol- 2 n operator [ string formulation of mensional theory; they appear to tudes take the form given in [ litudes naturally take the form [ es from which the ABJ(M) theories -ordered amplitudes. For four-point ]. s all factorization properties [ 36 , ] shares many of the properties of 35 16 , = 6 conformal supergravity was discussed from a field – 2 – 26 N ] have been discussed extensively. Another interesting 33 – ] as well as a formulation [ 31 ]. The ABJ(M) amplitude expression of [ Unlike its four-dimensional counterpart, three-dimensio 22 24 3 = 6 Chern-Simons theory (ABJ(M)) as a subsector to which it ca ]. ]. We shall also discuss the consequences of conformal super ]. It was suggested that the scattering amplitudes of this th 26 30 N 21 = 6 superconformal field theories that are not immediately in – ) N 19 M ]. 34 SU( × ) Generalizations of the ABJ(M) theory, to more general gauge The crucial difference between the twistor space of the three In this paper we discuss the construction of a (twistor) stri The three-dimensional ABJ(M) theory [ N It may also be possible to construct a string theory whose amp The analogous relation between the Grassmannian and twistor The coupling of the ABJ(M) theory with 4 2 3 level amplitudes was discussed in [ be consistently truncated; the resulting scattering ampli the classification [ equivalent to an alternative integral formula which satisfie the lines of [ only an orthogonal one.level The of orthogonality constraint the may worldsheetat in action tree-level at it the mayscattering also expense states. be of imposed introducing More only precisely, as the a string choice theory of we the sha kin Minkowski spaces is that, while the latter has a unitary symm SU( lower numbers of supercharges [ question is whether there exist higher-dimensionalcan theori be obtained by some truncation procedure, such as dimens twistor string theory may exist fortain this conformal theory supergravity. as well.conformal Such supergravity does notlead have to factorization any channels asymptotic whichits are states not presence present can in only thewhich fl be vanish inferred in the either absence from of off-shell conformal data supergravity o inter of its tree-level amplitudesformal and group of [ its loopa integrands Grassmannian unde formulation [ calization on curves of a certain degree in twistor space, sYM theory such as the integrability of the planar dilatatio and some of itsalso conservation turns out laws that, without withinof identifying our other the construction, consistent co truncations it iscorrespond of possible this to putative higher-di While this string theorysentations appears of to a have larger infinitely symmetrydimensional group many field and stat theory. thus may This be construction interpr yields the scat interactions: they generate tree-levelamplitudes multi-trace color we compare explicitlylowing the from string-theory-gener the Lagrangian put forth in [ theory perspective in [ JHEP06(2014)088 ), N . In ]. In 5 | 4 23 ) where A ntributions , η , one finds a µ we construct ξ m 4 = 4 sYM theory. es to amplitudes and N with remarks on various gular or vanishing factors ll as the general structure through holomorphic delta 6 heory the only contributing tex operators, as suggested tions of Sp(4) and SO( a subset of states which are n two factors, the inclusion ompact form of) CP ; (2.1) th vanishing charge under a ]. In section mparing them with the ones nerators are Grassmann-even ectrum at the quantum level. e third component of the mo- ld theory, restricting to three- g to states transforming only over the expression of [ 13 and its states and discuss the AB vel scattering amplitudes of the [ leaving the states invariant and of our construction as well as on ction δ 5 | 4 = } , loop amplitudes of chargeless states receive with some fixed matrix B we discuss an embedding of the twistor uss loop amplitudes in this paper. A α 2 , η q A α A η ]. We also discuss a similar comparison for { m ). Similarly, if the states are invariant under ]; they are given by the pairs ( 26 3 i = – 3 – 39 P Q , =1 n i µν )). Thus, to extract the scattering amplitudes of such P ( A α -extended three-dimensional superconformal group δ ) i q N ] = Ω ≡ ( ν α A , ξ m (0) µ δ ξ [ =1 n i P ( δ For general states, the amplitudes of this theory receive co in close similarity with the twistor string for the 6 5 ], (0) = δ 37 ) were discussed in detail in [ , 3 we construct an open string theory on CP R we discuss the contribution of conformal supergravity stat ] for a detailed discussion on holomorphic delta functions. , 4 3 5 38 ]. The localization on instantons of fixed degree is realized N | As in all theories with manifest symmetries, restriction to This paper is organized as follows. In section 23 While this truncation is natural and consistent at tree level See [ 5 6 respectively. They are real and self-conjugate, the two components transform in the fundamental representa 2 An embedding ofThe the 3d twistor supertwistors space OSp( for the certain arbitrary-multiplicity amplitudes.extensions We close of in our construction se of to other gauge states groups and with thethe more higher-dimensional possibility tha interpretation of enforcing the truncation to the ABJ(M) sp the scattering amplitudes ofsection the constrained states andillustrate rec it by computingfollowing the from four-point the Lagrangian amplitudes proposed and in co [ space of the three-dimensional Minkowski space into (a nonc by [ functions [ from instantonsunder of the all expected SU(3) degrees; on-shell R-symmetryinstantons of upon are the ABJ(M) restrictin those t of degree equal to half the number of ver contributions from charged states as well. We shall not disc of scattering amplitudes of this theory, following closely truncation of its space of states to the ABJ(M) spectrum as we section dimensional kinematics specified bymentum e.g. yields the a vanishing factor of of th invariant under a subgroupin of scattering the amplitudes, symmetry depending groupor on leads odd, whether to respectively. the sin inert For example, ge in a four-dimensional fie certain worldsheet global U(1) symmetryABJ(M) yields theory. the tree-le symmetry and restricting to a particular subset of states wi some on-shell generator extract the corresponding vanishing or singular factors. invariant states it is necessary to identify the generators factor of 0 = JHEP06(2014)088 . ) 8 . λ 2 are , 2 z, y (2.5) (2.6) (2.4) (2.3) (2.2) µ . The identi- a ons are omplex λ and λ , ) , , λ I ( K y ¯ φ η ). J K η ) embed this space as η I 2.9 ]. Unlike the twistor J η 0. It is however not clear , η s construction; we shall ) 2.1 39 . 2 I y with target space ( λ , η ) ( ˙ a e coordinates, e.g. ic state kinematics. While n > ¯ λ ). When restricting the extental = 1 s a phase space on which one sonic coordinates and append , onstraint on (wave) functions ) + IJK ies of spinors make it difficult α ˙ λ 2.2 a ψ ( µ with K , + 2 ) which implies that ¯ ψ n, = (¯ J 3) J η 2). The three-dimensional conformal 7 I ) as part of the coordinates of CP 2+ of eq. ( 2.2 η I , b a, I η ( ) such that . ) η . , µ b is proportional to the canonical conjugate of 2), acts linearly on the first two and the λ ive coefficient, see eq. ( a , z ≡ , ( ab µ λ , µ ǫ 3 ) + ) a IJ . = 0 , I λ the canonical conjugates of λ β φ 2 y ( a = ( λ ( , SU(3 ] = I ¯ µ α b – 4 – + µ f z ˙ 2) symmetry, vertex operators for the restricted a JK = ( λ ξ ⊂ I , I ¯ µ φ = 1 , µ y µ η = I I a ) ξ ˙ aa 2) η λ iz λ ǫ , [ αβ ( 9 I P y e , α direction is treated separately. ) + ψ : 5 λ λ d 3 ) SU(2 ( ˙ a µ ) + Z , simply enforces eq. ( ⊂ , µ λ 2 IJK ( ) α Z ¯ φ ψ λ R ) = . We denote by , I I z y . This identification, which breaks scale invariance of the c ), the regular twistor space is obtained as the hypersurface = ( ( 3 ) = ) = 4) are second-order differential operators [ ˜ f I µ y = 6, for which the two types of supertwistor space wave functi y, z N | λ, η λ, η ( N ˆ Φ( ) is homogeneous it only implies that IJK , i.e. as a representative of the solutions to the constraint ˆ Ψ 2.5 µ and ¯ µ We interpret the bosonic coordinates From a bosonic point of view we shall construct a string theor To have more manifest symmetry we may relax the constraint ( It is not difficult to include the Grassmann coordinates in thi E.g. momenta are still bilinears in Since eq. ( One may in principle consider larger spaces, such as CP 7 9 8 projective space to only a located at a point in ¯ fying In the phase space ( canonically conjugate to each other. This relative coefficient then appears on the right-hand side group, embedded as Sp(4 states we shall pick the particular solution with unit relat space of the four-dimensionalto Minkowski linearize space, the the action propert of the (super)conformal group. a hypersurface in a largerdefined one; on we this subsequently larger space. imposeafterwards this Let the c us begin Grassmann by directions. discussing the bo last two components of transform in the fundamental representation of SU(3 The homogeneous coordinates, may choose Darboux-like coordinates, From this perspective the twistor space may be interpreted a whether this will make later considerations more natural. Thus, (wave) functions on this space depend only on half of th states do not as the additional ¯ generators of OSp( focus on the case and impose the identificationthis as string theory a has specific the choice complete of SU(3 asymptot JHEP06(2014)088 z ⊂ = 6 1) (2.9) (2.7) (2.8) , (2.10) (2.11) N and choose ); however, 1 , 4 and that of ) we may con- 2.7 | 2 , η 2 2.5 1). Then, we break by , 4 y sted in the coefficient tion is quite singular. | 1) with an SU(1 , 2 , . 3 t of all external momenta. space CP ) y break it while preserving than in eq. ( led to the way it is broken nce the bosonic part of the • SU(4 ). We will do this by applying ) as well as the non-manifest )( s in terms of the coordinates 4) symmetry by imposing the f the three-dimensional | 5 ⊂ rdinates, such as the vanishing , ¯ η . 2.8 ˙ 2  1) 2 µ , ia µ ¯ µ + ˙ . a = ¯ , i 4 ). It turns out that for our purpose it ˙ ¯ 2 µ η ˙ 1) ) in the form )-invariant constraint ( 1 ˙ , aa 1)-invariant supercharge, 2.5 = 0 µ R ǫ , ( , 2.5 5 δ ¯ η i 5 ˙ 2 ¯ η X , µ SU(4 µ d – 5 –  4 1 ⊂ + δ ¯ η µ d 4 ≡ ¯ η = ¯ ˙ 1 Z ˙ 1 µ (0) µ SU(4) δ ) = • ( F P , they are Fourier-conjugates of each other. To this end we y and 3. We will denote the fermionic completion of z , 2 , to some value and impose eq. ( 3 = 1 µ 1) to its SU(3) subgroup by identifying SU(1 , ) and imposing the vanishing of an SU(1 ). We then obtain the states of a theory with OSp(6 ¯ where the sum runsof over such all a external singular states. factor. We will be intere As discussed in theIgnoring fermions, Introduction, it is such akin a toIn setting kinematic the to zero configura case one at componen hand we expect the singular factor to the result of item 1) the projector η R I, J, K ; similarly to , To summarize, we shall construct a string theory with target η 1. We fix ¯ 2. Impose the vanishing of the supercharge component ( µ, µ, following constraints on the external state kinematics: to express the kinematic(¯ information of the vertex operator on-shell superspace appear to be less natural. constraints breaking it to the requisite SU(3) R-symmetry o sider a similar constraint involvingof the a Grassmann-odd supermomentum coo componentis similar useful to to eq. choose ( such that the complete symmetry of the embedding space is SU( with the SU(4 by ¯ Sp(4 supertwistor space is obtained through an SL(2 should embed SO(6) into athe larger manifest symmetry SU(3) group on-shell andSO(6) symmetry partiall symmetry (of (of asymptotic amplitudes). states and the The wave suitable functions choice on is the coup twistor space are recovered. Si It is in principle possible to choose a symmetry group larger JHEP06(2014)088 . = N =0 5 that , ], an A (3.2) (3.1) (3.3) 4 (2.12) ¯ η 13 | 2, . , G 5  ¯ η ∂ = 1 ,jR , 2 ˙ a Ψ )) , and 3 L ρ ) and the coordi- ( , ∂ =0 2 nd that we should A 5 3.1 ,R , , ing of a supercharge j 2 4 ]. , ψ ¯ η he U(1) that rephases ¯ | ) Ψ mposing the vanishing 1) symmetry, explicitly n. The only possibility s due to its Grassmann 40 ρ , = 1 manifestation — though eflection of the fact that G . ( + 4 ˙ a ¯ η α G edom. They consist of ∂ S , µ ,jL ) 2 w construct, following [ ρ 1) ( Ψ external state kinematics. To avoid ( , ) + R α 4 . I R | with a standard action ∂ λ 2 Z ia 2 ,L ) which eliminates the constrained , g the lines of [ L j 2 ¯ µ ˙ a ¯ Ψ ∇ i ) = ( 2.11 11 µ ρ RI ( =1 M ˙ j I aa X Y heet position as argument of the worldsheet fields. ǫ + + contains the connection for the local GL(1) 2 fields Ψ i restores this symmetry, albeit non-manifestly. I L / X 5 ,iR , ∇ Z – 6 – 1 4 ,Y R which relates the free fields ( η → (0) whose argument is in fact the corresponding Ψ δ ∇ and its truncation to ABJ(M) states L )) Y ∂ ρ 1 (0) LI 1 ( , δ − ,R Y A 1) respectively and have a standard first-order action 4 t i 1 | ( , ¯ ) acts manifestly on functions on the three-dimensional ψ 2 ¯ and its vertex operators that correspond to the ABJ(M) , ρ Ψ , R → 2 2 dimension-1 ) 1 ) is a particular combination of , , d + ρ 4 ( | Y G ˙ 2 M a ( , Z , ,iL ¯ 2 λ F 1 Sp(4 , = tZ P . While this projector preserves an SU(1 ) Ψ ) is the action for a collection of two-dimensional fermions 5 , and ρ ⊂ R S ( η → G 1 ∂ ) 3.2 α on CP R µ ,L Z i 1 , and ¯ in ( 10 ¯ Ψ . 4 1  G η , ) = (¯ ], the covariant derivative 4 ρ S | ( 2 N =1 13 , 2 fields Ψ i I X 2 / Z ρ 2 d Z 5.). Their dimensions are (0 solving the constraint imposed byonly it one will SL(2 break it; this is a r twistor space. The integration over ¯ Similarly to the bosonicof constraint, a we component shouldfind of expect a the that Grassmann supercharge i delta is function a rather singular limit a The functions produced byuniformly this ¯ projector are charged under t components through the integral operator ( component of the supercharge;nature. such a Unlike delta the function bosonic vanishe case however we impose the vanish We shall see that this expectation is indeed realized. Grassmann variables; we should thereforesimilar expect in a different spirit —that of is the independent vanishing of Grassmann the delta number functio of vertex operators is Given a function = The term We shall use the same notation for the worldsheet fields as for Presumably, a heterotic theory can also be constructed alon G 11 10 ,..., S Similarly to [ 1 potential confusion, we shall include explicitly the worlds transform in the fundamental representation of SU(3 symmetry acting as nates on CP will be responsible for the target space gauge degrees of fre open string theory 3 A string theory on CP With the ingredients discussed in the previous section we no states. The worldsheet fields and their conjugates, dimension-1 JHEP06(2014)088 . )- R Z M (3.7) (3.8) (3.9) (3.6) (3.4) (3.5) = SU( L ) is also Z . × ) 3.3 I L = Z N Z . As usual, the LI 2 o the discussion Y , n be gauged. The and GL(1) ghosts ) and ( = )) the ghost systems is ρ pace effective theory. and Ψ ( 3.2 b, c A 1 ¯ GL(1) ψ ξ  ) decomposition. rent are c , − nishing correlation functions onal twistor space (and thus which we call M nctions are SU( L undamental representation of A i,R 12 e can pick any desired numbers η ,J ave no anomalous term in their SU( cb∂ (¯  ntly on Ψ 5 2 L | × + 0 = Ψ -closure and dimension constraints. ) Ψ δ ∂v , the action ( 2 ) N Q ρ L ¯ Z i,L ( Ψ u , ˙ a Ψ GL(1) ¯ j λ 1 M ξ ˙ Ψ a vJ ) + i and a , − L iµ + T c R 1 e i Y ) and the Ψ-dependent currents written here generate L Z Ψ ¯ )) b Ψ uv – 7 – M 1 -independent part of vertex operators have the ( ρ i = ( ¯ L cT = Ψ + ¯ µ ∂ L )) differently) the theory — and its amplitudes — 1 a ξ ij + N N ρ + (  − ˙ 3 L cT q ,Z µ  ,J (¯ , µ ∂c = R 3 ) ij L δ ρ Y dρ b J F 1). The Ψ ( ) , ξ 3 J = dξ ρ + Z 4 λ | ( ], which is determined by generate SU( L 2 G = U Z a , Y ) yields the usual diffeomorphism ghosts T 41 T , Q = + 3.2 ) = 13 I L U ρ Z ( L ij V ∇ LI Y 28 and should be cancelled by the fermion system. Similarly t − = 1) guarantees that this current is non-anomalous and thus ca ) system has vanishing central charge; the central charge of ], this restricts the possible gauge symmetry of the target s , 0 4 T | 15 2 = of fermions. 3 , Y,Z , − Before choosing external states to lie in the three-dimensi Apart from the symmetries acting on On this theory we impose open-string boundary conditions, Quantization of ( ; the corresponding left-moving stress tensor and GL(1) cur The numerical matrices 13 M / 12 26 same structure as in [ are invariant under SU(3 choosing to treat the pair ( and thus vertex operators depend on a single set of variables traceless part of theOPE corresponding with left-moving the stress currents tensor h invariant. and Of thus imply the that remaining correlation U(1) fu currents, their difference invariant under unitary transformations acting independe Classically however this cancellation isN unimportant and w also has vanishing mixed anomaly andhave thus vanishing require charge. that nonva the same algebra; for later purposes we write it here in an SU( u, v − The equal number of bosonicSU(2 and fermionic directions in the f The ( BRST operator takes the usual form: in [ The ones to be dressed with various combinations of Ψ are JHEP06(2014)088 e ) and (3.15) (3.13) (3.10) (3.11) 3 ¯ µ , , = 4 sYM 3 λ )) ; (3.12) ρ N ( i ) A ¯ n stanton number. ψ ρ ξ ( n − 15 V . n A ) η ]. (¯ ρ dρ 5 ( 13 | e tree-level the standard construction, I 0 interesting to analyze their theory. Z . δ be a sector of gauge-singlet ) ) ρ ∂Z sily be written down, it is not ( ding to the pair ( solve the constraints ... ρ ˙ a ( )) ) ), etc. [ I ¯ λ ρ tates describe conformal supergravity; 4 A ρ ξ g ( gh appropriate Wick rotations. ( ρ ˙ a ψ 1 ( Kaluza-Klein states. It also implies , Z ¯ A 4 iµ µ ( 3)) and introduces a few more possi- / ¯ e | V I ψ ) and 2 4 g ρ rmal supergravity-like states” even though a )) = 0 , ( I ρ 5 =4 2 dρ = I ( f X A µ ; (3.14) ¯ f µ g I F Z ξ V F q ∂ ) q 3 − ρ µ ( – 8 – ) + = 2 (¯ 3 ρ 2 ( δ ) it makes sense to consider a current constructed B ) cV 3 q n ) ρ λ = 0 ( 2 3 3.4 )) I ρ I ¯ ρ µ ( Y g ( 2 I B 3 in terms of ratios ¯ )) 2) allows us to construct a family of dimension-zero q ¯ µ Z , ρ . cV non-vanishing correlation functions have vanishing charg ′ U ( ξ ) ( 1 J Z F 1) symmetry (to SU(2 ) = ρ ( ξ q , ( ρ dξ I 1 4 ( | ′ f 2 Z J cV , h = and f ]. B ) = and express V 2) to SU(2 q ) = ρ 37 , ξ ; , ¯ η , 15 ˙ . . . n a (1 , µ a n 14 are primary operators if the functions µ A ], they break up into a sum over sectors labeled by the GL(1) in (¯ g n are 13 V U ) breaks the SU(3 5 13 Breaking SU(3 The scattering amplitudes of this string theory are given by The strange signature of the target space can be undone throu One can eliminate In theories for which a target space Lagrangian is known such s , ψ and with respect to the current 4 15 14 13 f ′ ψ q as in the case of the similar open string theory describing th Its mixed anomaly vanishes if for such a choice of ( operators from these fields, While the complete tree-level S-matrixclear of what this quantum field theory theory canstructure it ea and corresponds understand to. whether It or would not be it is a standard field bilities for vertex operators, akin tothat, vertex apart operators for from the GL(1) current ( V in the following wetarget shall space generically Lagrangian may refer not to be them known. as “confo as discussed in [ 3.1 Vertex operators withTreating reduced differently symmetry the target space kinematics correspon states while those that do not require such dressing and thus descri theory [ JHEP06(2014)088 1 , − ) and ann U ρ ; ) and 1 (3.20) (3.16) (3.19) (3.18) (3.17) ¯ η = 0: − , . . While ˙ a 3 1 U ) 3.13 µ . ˙ − F 2 . . defined in ), they are , µ η µ U P ¯ a ψ ) F )) a ij µ q ρ ρ (¯ J ( ( F 0 − 5 η, ξ and ¯ q A (¯ ¯ U ψ ¯ ein modes of the ˙ ψ a = u and are analogous F ξ ξ µ i P F ) = , + q f − a ˙ 1 21 V osition of the operators urrents µ A + µ n Ψ J F f this current we list the η ) ) on the point ¯ defined in eq. ( (¯ B P ρ ; to justify it we examine ); for all of the operators 5 q ( e theory to only the states ′ ( | through 6 ′ 4 0 ) = 3.9 J − q ending on three Grassmann ¯ )). To project the fermionic δ ρ ψ A ) equal number of t ρ ; twisted sector vertex operator 2.11 on ξ η ρ ( ¯ ( η ( ˙ a ) = ) = , A δ s interpretation here. ˙ ¯ δ λ g a ¯ ψ ξ )) function. Thus, we may trade V U ˙ ξ a ρ , µ , F δ , ( F which is taken as above. iµ i a 0 − A P e P , µ 3 . ( U ( ¯ (¯ ′ ψ ′ A )) µ F F q ξ q F η ρ 21 q , (¯ ( P J 0 5 − 3 | V M ¯ 0 µ + A δ ξ ′ η ( + and F (¯ J ) δ NM q 3 – 9 – 1 ρ | N ; such an orbifold would project out )) ; 0 = − introduced in eq. ( 3 ¯ ρ δ η − U ¯ ( µ , + 1) F ˙ ¯ a µ F = U(1) P n P − J , µ F ) with respect to the bosonic part of the current in q )) = q a 7→ − ρ B µ ( µ q (¯ 3 (¯ = (2 2 A µ 1 − δ ¯ ψ : ¯ − F )) = ξ ξ q 2 U ¯ dξ charge for the fermions Ψ ψ Z F − + q Z P A ). The external kinematics is specified by ¯ η, ξ B η (¯ (¯ 12 u 3.9 nq 5 J ) = ( | ′ 1) symmetry is unbroken at this stage, the target space Grassm 0 . ρ ) apart for the dependence on ¯ q ′ , δ ; ) = J ¯ ) = η F ρ , n P ˙ ; 3.10 a U ¯ η enter only in the combination , F , µ ) in eq. ( ˙ a a ) = η P ρ ( ¯ µ ( ), using a limit representation for the Dirac ψ ′ of the vertex operators thus constructed: , µ (¯ q a U δ ′ orbifold acting as ) and may be interpreted e.g. as a (singular) limit of a superp q µ ). These operators may be interpreted as being the Kaluza-Kl U η, ξ (¯ 2 3.15 (¯ ; it turns out that it is possible to consistently truncate th Z u 12 3.9 0 , In addition, there are vertex operators which do not depend o A further vertex operator may be obtained by localizing ( To understand some of the consequences of the conservation o Since the SU(4 3.13 1 U − F V to those in eq. ( eq. ( This operator carries charge ( described by them. We shall revisit this truncation in secti the new current thus, since after projection all vertex operators depend on above their fermionic part becomes in eq. ( operator We notice that it carries nontrivial charge under the curren the appropriately-dressed operators eq. ( which are to be dressed with the Ψ-dependent dimension-one c all charged under P For a particular choice of potentially interesting and useful, we shall not pursue thi for this operator. Itfor may also a be possible to interpret it as a coordinates kinematics onto the desiredvariables one we corresponding act with to the states projector dep An interesting class of correlators are those containing an charges JHEP06(2014)088 ). Z 1, 3.20 (3.22) (3.24) (3.21) ). The − ) can be effective k ; (3.23) 2.6 n,k 1 A 3.20 − ) is factors, their ) m i =1 1 ∞ m ρ . k X A = 4 open string ( − ) k U m , z i = A ( N ) ) to the states ( z i n m ) ˙ a 2 ( n and =1 ρ k 2 ( , z X m 0 ρ ) 0 ) introduce the alternate ( have chosen the perhaps U V m 0 ), enforcing the fact that (M) multiplets ( ex operators ( α ( n s; thus, the former can be V z ut rather include it in their 2 ) = fields transforming nontriv- his choice makes it straight- n ums over sectors labeled by i 3.20 m disk correlation functions 2.9 h instanton number 2 ase of the ρ ction of chargeless operators dρ half the coordinates and with ( on the possibility of truncating at tree level erators that can be constructed dρ A y. variable in each vertex operator 6 Z te charges and thus no chargeless se zero modes. The expression of ¯ ψ 1) symmetry, due to the presence Z is consistent with the expectation ξ , in eq. ( te it i ... 4 | ... 0 , ) 2 ) 4 V . , 4 1 ρ ρ ( − ( • 0 0 m i V and ) ρ V 4 5 ) 1 4 | m 5 ( − m dρ ˙ a ( dρ V z dz – 10 – Z Z ) ) =1 k =1 k collective coordinates ( ) with SU(3 3 3 Y X m m ρ ρ k ( ( 3.9 1 1 Z − − ) = i ρ cV cV ( ) ) ˙ a 2 2 ¯ ρ ρ λ ], ( ( 0 0 41 , cV cV , in the definition of ) ) 1 1 1 13 − 21 ρ ρ ( ( m i J 1 1 ρ charge. Therefore, from the perspective of the target space ) − − m α ( and cV cV z h h U(1) 12 q = =1 J k )= X n m n,k 2 2 A ) = ... i zero modes; since there is no non-trivial OPE between the ρ (1 ( We also note that the chargeless states described by the vert k n α 2 ¯ at the position of each vertex operators is, µ A as well as in the case of amplitudes ( has The currents correlation function is solely givenZ by an integral over the and the integration measure over the effectively imposes a gauge-fixing of GL(1). In the sector wit ordering used here. The integration over the auxiliary correlation functions. 3.2 The generalColor-dressed structure amplitudes of are scattering computed amplitudes of in integrated the and usual unintegrated way, vertex fro operators. As in the c the twistor space is a phase space. We shall not do this here, b consistently truncated away. We shall commentto in chargeless section states at higher orders in perturbation theor put in one to oneabsence correspondence of with vertex the operators states that of the are two independent ABJ of Ψ Indeed, introducing chargedrequires operators in fact in the introduction a ofsource correlation two can operators fun of appear opposi in the equations of motion for charged field ially under the gaugeunorthodox choice group. of labeling Tothe the construct momenta external these conjugate kinematics to operators with theforward we other to half of impose the on coordinates. these T operators the constraint in eq. ( that the only asymptotic states of this theory correspond to of the GL(1) gaugethe field, instanton correlation number [ functions are given by s action for this theory it is possible to consistently trunca have nonzero have vanishing charge with respect to it. All other vertex op JHEP06(2014)088 ] o 21 in ) n 2 n a 2 (4.1) n J (3.26) ...T ,..., alternates and +1 l 2 21 +1 a 12 l J T ree-dimensional etc. n J and ]Tr[ l ). They determine )cyc(2 2 ) l l a 12 of the instanton and 2 − J 3.9 n ( k string theory 2( s up to cyclicity: ...T tering amplitudes of fields ij Z ,..., 1 the arguments of the delta 1 J , ested mainly in the vertex a 4 × rgravity. In the case of the ceive contributions from the | ) and evaluate the integrals singularities of the integrand. T l n y part of the vertex operators 2 ify the integration contour; we , forming nontrivially under the 2 2 th a single-trace color structure 2 cyc(1 Z Z antons with degree Tr[ 3.20 / / tures necessarily comes only from   l 2 . ) ) S 1 ) which correspond to the directions X n n ρ − 2 2 1 ( V ˙ − a =1 l X n ¯ λ ) (3.25) + ,..., ,..., ] 4 4 +1 , , i n . n 2 ρ constrains the degree ) ) and a (2 (2 = ρ ) of the projection operator. For a fixed number – 11 – n − S S 5 ..., ( , 2 k i 3 4 × × ρ ¯ µ ...T ψ ( such delta functions. Since they are linear in the 3.16 1 2 l 1) 1) ), ¯ a − ,..., n = ρ i Y T ( m − − ] that the entire tree-level S matrix has only the former 1 5 is used instead of ) a l n n ¯ ψ 19 δ 2 2 ρ T cyc(1 V ), − , the insertion points of the operators ρ Tr[ ( ij for each one of the two fermionic directions), the only nonzer m 4 δ ,..., ,..., factors is trivial, albeit somewhat cumbersome to write out 1 ρ ¯ 3 3 ψ k +triple-traces + S , , X 22 i ∝ = (1 (1 J j i S S ) = ( ) Ψ n i m 2 ρ and Ψ = = ( h ), so we sketch here the structure of the correlator of l 2 1 n 2 11 S S a 21 J , . . . , ρ that are orthogonal to the hypersurface representing the th l 3.20 ρ 1 , 4 ...J | 2 ) , cyc( 1 2 fermionic moduli ( ρ ( The integral over the moduli of A common part to all correlation functions determining scat k of vertex operators there are 2 1 a 12 × n J h containing these moduli is the result ( relates it to the number of vertex operators. Indeed, the onl Inclusion of where the sums run over the permutations of even and odd label twistor space. of CP currents. Since general. While from a field theoryare perspective expected amplitudes to wi receivegauge contributions only group, from the fields amplitudesexchange trans with of a gauge-singlets, multi-trace suchABJ(M) structure theory as re it perhaps was conformal argued [ supe 2 charged under the gauge group is the correlator of currents on the boundary of the disk. Thus, 2 over moduli of the fields color structure; thus, allsuch contribution singlet to states. the latter struc 4 ABJ(M) amplitudes from the truncated CP the various trace structuresoperators of ( amplitudes. We will be inter Let us now restrict ourselves to the vertex operators ( contribution to their correlation function comes from inst will take it suchfunctions that and it requires instructs use us ofA to Cauchy’s similar theorem sum contour for over is the the other chosen zeroes if of To completely define the integral it is necessary to also spec JHEP06(2014)088 ¯ λ i 45 ct ) I n , 2 (4.4) (4.3) (4.5) (4.2) i at for ρ ) ( n 2 21 ρ to enclose ). In the ( . We shall ) . 21 ...J m exp ) 3 ( ) ) I 3.25 z m 1 16 3 m ( 3 ( ρ ...J has the same value. z ( ) dz δ 1 12 V ρ . Since there are as J ( =1 k . 1 h Y m − ) 12 for )) equences of the special 1 i J V 1 moduli is − h ρ Z − ( i m 3 ) integrals over the variables ) V o this end we parametrize ρ j A 1 µ i is given by ( ρ ¯ ξ ψ − ( ( i i 1 on contour for } 0 m i ξ ) δ ates of the other half) while contains the integral over the ρ U n V trade . ) is trivial because the only de- i 2 − } k ξ ρ . ∈{ ρ 0 ( factors in i ( ( A i V exp } 3 1 ) ) (that half of the three-dimensional η I 1 1 Y 1 21 µ (¯ det ∈{ − stand for the set of labels of operators − j 3 − V | 2.9 ) 0 } )) m i moduli in the presence of the kinematic i i δ ∈{ ρ ...J i ρ ρ 5 )) = i ( n i ( ) ¯ ξ =1 2 ψ 1 ρ 3 1 i Y ,... ( ( det ¯ − } ρ µ 6 1 3 5 , i 1 ( , ¯ µ U ) moduli, 4 ξ − i 4 = I } 12 ρ ξ and V , 1 ( – 12 – ( J ) 2 δ 4 3 − i h )) we can simply change variables to decouple the ∈{ exp } ¯ i V µ ρ ψ I δ Y ( V 3 1 ∈{ 3 4.1 I i Y det ¯ ∈{ µ ) ) } ≡ { i i 0 ξ ) m m = I I ( ( V m } 3 ( 3 { 1 dz I dz − moduli parametrizing ¯ dz 3 5 | | V Y 2 5 ) =1 k and ). As discussed in the Introduction and in section 2, to extra d d ∈{ m Y i m 3 ( } ) z =1 k =1 2.9 k Z Y m Y m m 3 ( i = i ,... matrix in the denominator is non-singular. We also notice th i with ξ ′ 3 dz 5 dξ over , respectively, and I moduli space. Thus, the integral over the ¯ , k 0 is replaced with dρ i 3 3 =1 3 k V , 3 I × Y µ I m n 1 dρ n,n =1 2 i 2 k ) as well as of the constraints generated in the evaluation of Z =1 A Q and n i Vol GL(2) -point amplitude is = 2.9 n } ≡ { 1 k,n Q Vol GL(2) 1 Z δ 3 − contains the integral over the ¯ I − V Z V 3 = = I { = as The integral Next, we impose the kinematic restrictions and evaluate the To evaluate the remaining integrals we first examine the cons n,k n,n We note that the analogous integral appearing if we choose to 2 2 n,n n,n 2 16 2 A A A many moduli as such factors (cf. eq. ( constraint ( is the result of the integration over the pendence on these variables is due to the ( Indeed, in this case evaluate the various factors in this order. of type this choice of vertex operators we must choose the integrati In general, the the origin of the ¯ where that do not appearA in the three-dimensional twistor space. T kinematic configuration ( integrals: twistor space directions are to be interpreted as the conjug following we continue to denote the instanton degree by where moduli in the presence of the kinematic constraint ( Thus, the 2 JHEP06(2014)088 , lta 2 − (4.9) (4.7) l s (4.10) + ) which i m ρ ˙ a a ) ) appears δ one is not ) ˙ l a ˙ 2 ( i 2.10 ˆ z µ a i )2 2.10 . The integral ¯ µ m a ( z =1 n i + ) l P ˙ 1 ( ˆ z . ia )1 (which we set to zero ) ; (4.8) e ) ) ρ m i ) to a single (diagonal) ( ( ρ z ( 2 ] it is possible to extract ( da ˙ a ): ¯ 3.9 µ 3 ¯ λ variables. =1 i a 1 (1) ξ and account for the Jacobian 2.9 z ˙ a k m,l ( i e proportional to the volume moduli without imposing this off the singular and vanishing acobian factor from the change ) + e the origin of this singularity, iµ P Z e in [ ) of ( ) is imposed strictly before the ¯ ) ρ λ e 2 i ective coordinate i ( R ρ ˙ iξ )) 2 , ( i 2.9 e ˙ ¯ a λ ρ ˆ ¯ λ )) ( i i ξ ¯ µ 7→ ; (4.6) ˙ ρ a SL(2 i i ( ) ξ  ¯ µ ρ iµ × ˙ a i ( e i − ξ ) ˙ 2 i µ )) ¯ λ R − i a i µ , i (¯ ρ ¯ µ ( 2 µ (¯ δ ¯ – 13 – µ n 2 i =1 2 i X ) n δ ξ = 1 when it is zero. As mentioned earlier, the =1 2 ρ i Y n  ( l − ) =1 2 4 1 i Y i m δ ¯ µ ˙ a ( ) µ a (¯ z 2 2 ). If the constraint ( 2.9 δ d ) + . The second integral gives the delta function ( n eq. ( ρ =1 =1 2 −−−−→ k 2.10 i Y ( ˙ 1 Y (1) ) ˙ = 1 and m ) 1 i z ρ ¯ λ m ( ) the integrand is invariant under the shift transformation ˙ . a ( a ˙ Z a ) sets to zero the antisymmetric part of the argument of the de a ˆ ¯ λ 7→ i dz 2.9 ), respectively. It is not difficult to see that the factor ( = ′ ξ ) 2.9 ˙ to a i ! ρ ( iµ exp ˙ 1 2 ˙ 2.12 and the prime in the measure of the first integral signals that 1 =1 (1) unless ˙ ). I e ˙ Y ¯ a ) breaks the manifest SL(2 λ z )) ) l i =0 ˙ a ( =1 k 2.9 ρ 2.10 z ˙ Y ( 1 (1) m z ¯ µ | ) and ( i

= ¯ λ ξ Z ) − l = ˙ 2.10 a ( i = z ˆ ¯ µ λ (¯ 2 The relevant integral is In the first integral we can safely impose the projection ( To isolate the problem we trade the integral over one modulus exp δ I n =1 2 i Y restriction and carrying out manipulations similar to thos is responsible for the singularityof mentioned variables above from and the J of some integration variable.which may We be identified will as therefore the appearancerelating first of it a isolat to shift symmetry, ( function and thus itintegration yields over ( moduli is carried out it yields a divergenc quite naturally. Indeed, evaluating the integral over the an overall factor of factors ( a physically-meaningful amplitude it is necessary to strip identification ( Then, the constraint ( where ˆ where In the presence of eq. ( This symmetry allows one to set to zero oneusing of the the shift transformation) integration to an integral over the coll becomes supposed to integrate over JHEP06(2014)088 1 (1) : ) l 5 ( /z (4.16) (4.15) (4.13) (4.17) (4.14) (4.12) z . 4 (1) . n z e proper- ) )) 2 ˙ i a a and ρ δ = . ,..., ˙ ( a ) ˙ a a i l η ¯ µ 4 ( δ i =2 µ ) z i . l ξ a i ˙ a ( ¯ )) ) µ ˆ z , − ˙ 1 2 1) is not included. η a +1 are zero. The i ) ) i , − l a µ µ + m m i ) ( (¯ i anifest in the equation ˙ − 2 ρ 2 z i ρ +1) ) d we impose the strict l δ ( µ w m 5 ≡ − similar manipulations as ) 5 ) = (1 ( n i leads to a determinant ¯ =1 w z 2 ψ ) vanishes identically (since i Y ( ρ i ) l η nt is to be evaluated. Their ( z ξ ˙ l, a 2 ; (4.11) ( 5  5 ( . , s of =1 ˙ ˆ z a a ) 4 ¯ + ψ δ k m 2 pt I ,w 5 i )2 ˙ , l,a ˙ a ¯ 1 1 µ ξ i ( i m P η − ( for an integral over µ × =4 l µ ˙ + 2 z a i ) , in the presence of the kinematic w i A + -integrals yield only delta functions; i + ˙ | 1 ¯ µ µ i ) 4 ρ (1) + m w 5 ( z exp δ µ ) m ¯ n α 4 ψ ) l it is useful to change variables to + A ) I ( =1 ˙ 2 l 1 ( i ˙ i ¯ a X ( ψ 1 ) ˆ z ρ ˆ i z = det ∂z 7→ ( − ξ m  a )1 ˙ a ( 4 ) ( δ 5 m i ,w z m ˆ ¯ δ 2 and the pair ( ψ m ¯ ( ρ 1  ψ i , ( ) 1 , z n − ξ z l =1 2 − m ( ˙ a i Y a 4  ( + δ i w , – 14 – δ ˙ ) a = 1 z a l ρ m 1 5 δ ( n 2 i a ˙ δ =1 a ¯ 2 µ k i Y i and thus the =2 ξ a =1 , dz l η ) X ) µ ) k m l w n a i m 4 ( m =1 2 + ( 5 ( α i ¯ =1 X µ k P z 4 dz X is the integral over the fermionic moduli ˙ m 1 dz ¯  n i ψ δ =1 2 , . . . , k =1 k k i µ 1 X =1 = l X Y ( =1 k m n,k − i η =1 7→ 2 Y w k m ξ Y w = 1 2  4 Z ( ) α A l l δ ¯ 4 ) ∂ ( ψ a 1, 1 = det exp (1) dz dη I z det ) +1) − l l w k ˙ a =2 ) are linear in ( l Y = − k ˆ z Z 2 w 5 1 exp ∂α 1 ∂ , ( because the delta functions present in it are relevant for th × I z 4 dη z ). First we trade the integral over 4.10 I 1 (1) ( ) z = Z det 2.9 exp ) this integrand acquires the fermionic shift symmetry ,..., exp l,a 5 I , ( I exp 4 × 2.9 I I = 1 w = exp w and isolate the integration variable that disappears shoul I 5 ) which is manifest in the 3d twistor space. This breaking is m det , 4 R I = , exp becomes To carry out the integral over the moduli ˆ The last component of I exp exp I exp I it is a Grassmannfor integral). To extract this zero we carry out identification ( exponents in eq. ( Carrying out the integration over all fermionic moduli exce Thus, one modulus can be set to zero and consequently constraint ( I The entries of the matrix corresponding to unphysical value above; we shall use the shorthand notation ( the Jacobian of this transformation is SL(2 We included range is where we indicated the indices of the matrix whose determina ties of the integrand. We note that, similarly to JHEP06(2014)088 ), . iA  ˙ a a ! , η δ (4.21) (4.19) (4.18) 1 1 with ˙ a moduli ia i − . ) , slightly λ µ − w i 1 a i  m ) and it is ρ a ( ˙ − a a ¯ µ 2 m i z k i = ( δ ) and find ξ ˜ ely. We have ˙ ξ a 2.9 n i )) (4.20) i + =1 2 i i n i X 2.9 l µ ) =1 = 2 ρ i a i X ( n i  ¯ 2 µ = ¯ ξ µ

1 1 i ρ 1 ) to Λ z ( δ ξ w n are =1 g Jacobian factor: 1 A i i X 21  − ¯ η 3 − =1 , i k  Y w 2 a i µ ,...,k (¯ µ ) )), focusing on the single- ...J 1 1 (1) 2 ˙ 1 a a =1 ) z dex to δ δ − of the naive bosonic moduli can enforce ( 1 m ancellation is: ˙ ) a | ρ ) i n m i ) ( ral in the limit ( =1 2 1 l ly, defining the kinematic con- ρ i µ Y 2 e of variables ( i − a i 12 ,...,k z ξ ! ¯ w i 2 µ J ( 1. 1 h ρ =1 − forming (¯ m l 2 i − n m =1 − ) we find that the scattering ampli-  )) ξ × + i 2 | i w i ) ˙ ( a a 1 ρ n i } m i ρ δ 1 2 i − P 2 ( ρ ˙ × a ξ − i 4.20 A m i w V µ ¯ ρ k ln( ψ n =1 . a i ˙ l 2 i =1 X 2 ∈{ i d i X ¯ i ξ µ ,..., det 2  i ( µ ˙ R a ξ i a

; moreover the dependence on the n − ; ξ δ =1 2 δ ˙ ) and ( i ; a X and the one that appeared from the fermionic det A i i – 15 – 1 exp ) η I 1 = 1 µ −  =1 ,...,k (¯ 4.4 a i k m δ Y ,...,k 3 w I ( 2 | exp ¯ − µ =2  0 I ) =2 ˙ a k ), ( dz a δ 1 m n | 3 δ 2 =1 m 2 | − ) i )) ˙ | X a l 2 i i 1 i w 1 4.3 ( d ρ z ρ µ  − ( 2 2 a i i δ ¯ m i µ ξ ), ( ,..., =1 ¯ − n µ i  l ρ ( Y ξ ˙ i m ˙ a 1 a + n i δ 4.2 i × =1 2 i = 1 m i ˙ i − µ a X w i i ξ ρ i dξ ξ µ w  µ a i i (¯ k =1 × ¯ 0 l µ X | dρ det( 2 2 i and = det n δ ξ =1 2 i X =1 5  n 1 values as one fermionic delta function was treated separat n i n , exp =1 2 2 4 i  Y I I − Q Vol GL(2) det = 0 states in the string theory constructed in section × × n = exp 5 Z I exp ,..., , -integral produces a regularized zero and the correspondin 4 I η U(1) I = q factor cancels a similar factor in = = 2 exp takes 2 i 5 , I 1 n,n (1) i 4 2 z Putting together eqs. ( I A exp I reorganizing the integration variables and Fourier-trans trace component of the current correlator, making the chang cancels between the determinant in Upon dropping the factorsstraints on to the also third include line the (or, integral alternative operator tudes of parametrized by the sameintegral quantity in leading the to same the limit. divergence For the remaining determinant we integration. To see this one simply changes the summation in The where Here while the therefore extracted the origin of the zero value of the integ the appropriate upper bound on its range; the result of this c JHEP06(2014)088 ) , ) (5.2) (5.3) 4.21 ) and i, j (4.22) ]. ( 1 ] (5.1) − 36 n k 4.21 2 , 2 a ), breaking ≤ 35 ) Y , , i

T , p − 6 φ 1 δ 4) = p 34 14 = , ( c c e Chern-Simons vector ! 2 i , 34 12 32 p scuss first the case of the c c 2) 2; 3 , ts double-trace version can , l . For component 4 ) was computed in [ (2 =1 e couplings of the conformal = i X (1 ctures Tr[ A constraints, is lowered by the contributions ,C ,C 2 ) ) ,

t the entire super-amplitude is ter the factor ( , c i 4.22 3 13 11 2) 32 34 i i , δ c c 34 C C (2 3 h 31 34 i e amplitude; it will however be sufficient i h h 22 22 = = A 14 ) C C 34 − h 3 13 23 4 h η − − ) is = 2 4 = 2). Gauge fixing GL(2) as in [ − η 23 21 j 14 1 4 5.3 C C η 12 12 4) = )( ) we can extract its expression: 3 , 3 C C 3 η – 17 – , 2 )( )( 3 in the superamplitude above can be understood ] it is not hard to see that this amplitude should , c 5.8 2 η i i 11 13 i ,C ,C ( , = 1 and 26 1 2 ] superamplitude is C C i 21 31 (1 34 η 4 h h h 2 a 24 24 , i = 1 = 0 − η T C C 0) , 3 3 is the instanton degree that determines it and 12 22 a d = (4 − − T k A 4 32 =1 21 23 i i i Y C C ]Tr[ 34 14 Z 2 14 14 a ih ih C C T ) = ( ( , c 12 23 1 ¯ φ 4 i i h h a ) where ] (i.e. we choose = ,C ,C we will use for multi-trace color-stripped amplitudes is , p 4 T − n 32 31 a h h 2 23 12 12 14 T φ 3 c c F − 3 a 18 , p 4) = = = = T , ,..., 23 ¯ φ 2 11 21 12 (1 c C C 2; 3 ]Tr[ , p ,k , 2 ) φ 1 a p (1 ( ,... T 2 2 2 , 1 , a ,n Let us consider the four-scalar double-trace amplitude Inspecting the Lagrangian in ref. [ 2) 2) , 1 This notation does not uniquely specify a general multi-trac , T n (2 18 (2 ( A A Tr[ The presence of the high power of as being due toamplitudes having the other intermediate fields higher-derivative the action power of of the conforma Using the solution to the bosonic part of the delta-function of the fermionic delta function. is the easiest tosupergravity calculate and from ABJ(M) the fields. Lagrangian From and ( probes th be generated by a single Feynman graph with the exchange of th shown to agree with thebe ABJ(M) easily four-point reconstructed superamplitude. fromWe the I will details choose of to that consider solution the af single- and double-trace stru A for the purpose of our discussion. The notation The single-trace four-point superamplitude arising from ( the numbers of gaugefour-point group amplitudes; generators this in is eachdetermined the trace. by simplest a Let example, single us in component di tha amplitude. 5.1 The four-point amplitudes implies that the relevant multiplicative factor ( we find that the Tr[ JHEP06(2014)088 4) , 2; 3 (5.12) (5.10) (5.13) (5.11) itudes , (1 , 2 . , ve are the are SU(4) i 2) ) | B i . (2 31 ] and single- 123 4 A i ih 12 a ] and the fields 1 j 3 h , φ η µ T , 23 and 3 ) ˜ ollowing integrals 3 26 i h A 3 1 1 1 a j j η , φ A µ − 2 A 1 T 2 ˜ η interaction Feynman ψ A f amplitudes, is more (12)(34) (12)(34) = )( , φ A j − 3 1 c ]Tr[ 1 i η φ ∝ ∝ ) 2 φ 2 B 1 4 ing from the Lagrangian a itudes, determined by an ( ) ) η ψ − p ace component amplitudes 1 1 2 2 T i i B η 1 η η ∼ hat the other components of hoosing external states with A ( − A a (4) indices [ ) B ) tors are equal: i 3 mi mi T ) 19 4 η p 3 ¯ IJ C C φ ( d , IJ b (Σ 3 ) =1 =1 =1 ¯ 2 2 4 i 4 i φ 4 i (Σ ) coupling IJ p µ , 2 B i Q 2 P P B p φ ¯ + ( ( φ R 1 4 δ δ , 1 + ) ) 1 IJ + µ p 1 1 i i ¯ 1 φ ( i η η B ( p A ( 4 1 abc ψ mi mi ǫ  C C − a ... µ – 18 – ) , 2 A i + =1 =1 iB p 4 i 4 i φ ) 1 2 (12)(21)(34)(43) (12)(23)(34)(41) A i −  P P φ , φ + ( ( µ 1 µ are the SO(6) generators in the spinor representation. δ δ 12 p ab D iB γ A , φ =1 =1 i A 2 1 B ¯ φ 2 m 2 m ) µ ab 31 ) = ( µ + ) follows from the embedding of the manifest SU(3) on shell ω ) is it not difficult to find 4 Q Q , φ ¯ 4 1 µ IJ φ 4 2 2 4 4 eD ∂ content, the resulting single-trace and double-trace ampl 2.6 23 + , p  − 5.10 dη dη 3 φ 4 1 µ ( 1 2 ) follow from the same Lagrangian. 3 3 η φ 3 ∂ ] amplitudes, respectively. The denominators included abo = =  4 ∼ dη dη , p 5.8 a 1 1 1 2 2 i A ) L = ¯ φ 2 and T 4 i φ 3 dη dη µ A 3 3 , p a 2 1 1 1 , φ 4 are SO(6) indices in the fundamental representation, ψ 3 D T φ 1 µ , η dη dη 2 3 p 2 external states have the same expression. For example, the f a D J , φ ( 2 2 Z Z T , η , 1 3 1 (the SO(6) R-symmetry gauge field) in 2) a , φ η , 1 and T (2 φ IJ ( I µ A ] a different approach, which avoids the direct computation o different B Using the Chern-Simons propagator and the standard current It is also easy to check that the four-fermion amplitude 26 = 2 instanton, and notice that certain single- and double-tr 19 are equal. only differences between thethe two same superamplitudes; thus, by c The relation of the scalar fields carrying the scalar fields SU matching the result above.the Supersymmetry superamplitude then guarantees ( t fundamental indices and (Σ where trace Tr[ They represent contributions to different double-trace Tr[ in the on-shell multiplets ( R-symmetry group in SU(4): For a more systematicof [ comparison with the amplitudes follow 5.2 CGS interactions for any number of external legs field of ABJ(M) fermions with the same CSG R-symmetry gauge field. matches as well. This amplitude is determined by the minimal efficient. To thisk end let us examinewith again the four-point ampl rules following from eq. ( with different integration measures and integrand denomina JHEP06(2014)088 and 1 j (5.14) η )  2  C j and p φ Ai Ai 1 i l C ψ +  ψ   η  ¯ φ µ µ 1 j j γ B p A γ (4) j j B j (4) (4) (4) µ i µ i δ be the states that 2 µ i µ i 1 1 1 ˜ ˜ φ Ai Ai either contains the 2 1 A A = ˜ ˜ j ψ φ φ 2 2 ¯ φ ¯ A A l A ψ e- and double-trace ψ A i A i k p p q ¯ µ A j A j φ Aj Aj − µ φ φ (3) (3) (3) (3) φ φ ψ ψ µ µ + + D D 2 B j Bi 2 1 1 and ¯ ¯ ¯ 1 1 ¯ φ φ ψ D D φ ψ ψ i − − p p al difficulty is that, since l A i A i A e of the existence of similar ¯ Ak ¯ ¯ φ (2) φ φ (2) (2) (2) Ai = = · · · − · · · − Ai · · · − · · · − ¯ ( 1 double-trace amplitudes. In 2 1 ψ µ µ 1 ψ ψ q q ψ φ ψ ψ kl = = Bi µ = = µ s, as detailed below: ormal supergravity fields; while ij eD eD ); a judicious choice of external ψ D A i A i ion, let D (1) (1) hould be uniquely determined by (1) (1) Ai Ai − − µ µ φ φ 1 ef 2 1 2 Ak ψ ψ γ µ µ ¯ ¯ γ 2 ¯ ¯ φ ¯ ψ φ ψ 5.13 µ µ ψ 3 3 3 3 4 4 4 3 − Ai D D Ai kl D D η η η η ¯ ¯ ψ ψ ij 3 3 3 3 1 3 2 1 Tr Tr Tr Tr η η η η and any other state e 2 ef e 2 3 3 3 3 4 4 4 4 1 2 − − − − dη dη dη dη 2 j 3 3 3 3 3 3 3 3 η j i  dη dη dη dη – 19 –   3 3 3 3 2 2 2 2  ¯   and  φφ Bi Bi to contain the Grassmann variables Ai Ai dη dη dη dη B B ψ ψ ψ ψ j 3 3 3 3 1 1 1 1 (4) = 2, it is not obvious that simply interchanging two A A (4) (4) Contact term (4) +1 µ µ B B 2 i 2 Aj 1 1 1 dη dη dη dη γ γ A A k η ψ φ φ ψ φ ) ) (Σ) (Σ) R R R R and Ai Ai B Ai IJ IJ ¯ ¯ B i B i i (3) (3) (3) A ψ ψ (3) ¯ A i A i ψ ) 1 φ φ 2 1 2 µ µ φ φ ¯ ¯ (Σ (Σ ¯ ¯ φ φ ψ ψ µ µ ge IJ IJ D D µ µ ¯ φφ 1 8 IJ IJ µ µ ( D D B B − − B B 1 1 4 4 Tr Tr Tr Tr i A i A − Bi 1 1 4 4 ¯ ¯  φ φ    Ai Ai ψ B i µ µ 1 to contain ψ ψ + + φ Ai (2) µ µ (2) (2) (2) − ¯ 1 j A 1 2 1 ψ · · · − · · · − eD eD D D ¯ j φ ··· ··· ψ φ ψ ψ − − µ µ ge = = γ γ 1 4 Bj = = (1) (1) (1) (1) A i A i Exchange of spin-1 Chern-Simons field with Exchange of spin-1 Chern-Simons field with ψ 2 1 Ai Ai 2 1 φ φ ¯ ¯ Ai Ai ¯ ¯ ¯ ¯ φ ψ φ ψ ψ ψ Ai µ µ ψ ψ ¯ ψ µ µ + 1 and D D e e 2 2 i Tr Tr Tr Tr ge D D − − 8 3 Contact term and exchange of spin-1 Chern-Simons field with − The discussion above can be extended to a comparison of singl From a Lagrangian perspective this equality is a consequenc Grassmann coordinates has the same effect as in eq. ( amplitudes with any numberthe of instanton external degree is legs. larger The than one potenti states addresses this issue. As in the beginning of this sect This comparison probes only the matterit interactions does of not conf probe thesuperconformal self-interactions symmetry. of the latter, they s terms with and without conformal supergravity interaction the states are non-adjacent in thethe single-trace double-trace case but we are choose adjacent in the JHEP06(2014)088 and ) or 1 1 η e of two , ψ in their 1 ) implies 1 ¯ ψ η , 3.9 2 espondence of , ψ states and the 2 is a Calabi-Yau ¯ plitude the fields ψ 1 , 4 | n the form proposed 2 . Inspecting the field , 2 open string theory on reduction of a higher- IJ µ for these amplitudes we tion is the spin-1 Chern off-shell perspective and B up indices. Upon trunca- ent such asymptotic states ces can be mediated either some auxiliary fields interact elds litudes at tree level. By com- es several fields, including the e ABJ(M)-coupled conformal us, our construction suggests ) are either ( ices; this interaction is forbid- ble-trace structure for the field trivially under SU(4) the fields ). By explicitly evaluating the e shown that their interactions t to conclude that the interac- interesting to find a Lagrangian eory of the topological B-model e vertex operators ( 1 ed Chern-Simons-matter theories by , j appear together with 1 . Since CP j , φ ed from the equality of amplitudes ˙ a 2 − µ ¯ φ α , , j and 2 µ i , φ + 1 1 ¯ φ i, i . In the single-trace case we interchange the 1 2 – 20 – η ) or ( 2 , ψ and 1 ¯ ψ 1 , η 1 , ψ 2 , in close analogy to the ambitwistor space which is a subspac while leaving unchanged all the other dependence on ¯ 1 ψ , ) it is not difficult to see that they are equal. 4 | 2 +1 , i 2 η 5.13 -s implies that the fields ( and η j η ) are either ( . These interactions are the same as those leading to the corr µ or is independent of , j ) in the double-trace amplitude while in the single-trace am A 2 1 2 k η The spectrum of this string theory contains a finite number of − , φ 1 k reproduces the tree-level amplitudes of the ABJ(M) theory i 2 η ¯ φ , j 1 20 , , 4 ]. 1 | 2 The string theory also contains states carrying no gauge gro To figure out the gauge-singlet fields that can be responsible , + 1 It may be possible to extend this construction to mass-deform 23 , φ 2 1 . This choice of 20 2 ¯ i, i φ affect the S matrix byputing generating double-trace multi-trace amplitudes scattering in this amp string theory we hav tion to a three-dimensionaldisappear theory from with the the ABJ(M) spectrum. field cont They survive, however, from an space, this theory can also beon described this as space. the string field th 6 Outlook In this paperCP we have shown that a particular truncation of an four-point single- and double-trace amplitudes, asfollowing expect from our construction. integrals analogous to ( respective multiplets. Thus, apartsupergravity Lagrangian, from the contact only terms fields thatconfiguration in can described th yield above a are dou the spin-1 Chern-Simons fi and 2 indices of by a contact termSimons or field by an SU(4)-neutral field; the only such op configuration in the single-tracetion amplitude between it the is fields not which difficul belonged to the two different tra that it should conserve the six-component matrix ¯ follow the SU(4) indices. Sinceexchanged the between two them traces must transform carry non such(conformal) indices; graviton. this exclud Thethrough (conformal) vertices gravitino which as are well antisymmetricden as in because their of SU(4) our ind assumption that both particles ( considering two copies of CP η product copies of the super-twistor space. ( in [ truncation can be interpretedthat as the dimensional ABJ(M) reduction;dimensional theory (perhaps th non-unitary) can field theory. be Itinterpretation would interpreted of be the as latter; the the S dimensional matrix following from th JHEP06(2014)088 ]. = Q 42 F (6.2) (6.1) q s. For keeps w charges ) and charges to the current F ates in a color- q ) OPE, such that uged away except 2 nstanton sector re- ρ ( . m containing only the } . ) have variantized with respect N,N,N,N ions is that, in all trace 3) U(1) 2 32 while also cancelling the ]. It would be interesting , J q are inserted at alternating heory perspective. Such a ) ,J plitude are treated symmet- 43 (6 1 t gauge groups with specific 1 esting to understand if such pattern while preserving the , M M ρ tons of fixed degree; since a f the ABJ(M) theory coupled 43 color/kinematics duality [ − , this truncation is consistent ( 4) at this current be gauged. To 3 + V ,J , NM he “color instantons” to have the ons, this may be a possible expla- and 14 N U(1) (4 content and mirrors the fact that or J malous term in the OPE of the analog of 0 ,J − ij ion functions of the fermion currents N ∈ { V J 12 2 F ) q ,J ]. 21 = 2 J 26 N,M 2 ( q ) with multiplicities (5 4 – 21 – M charge, the gauge anomaly cancels only for Ψ → , 2 + , 3 NM 1 30. N Ψ − , 2 − = the currents ( Ψ 2 F NM , ). Nontrivial instantons will modify the fermion two-point q 1 ws 21 2 c ) = − 2), + 4.21 ) for the Ψ / 2 M B ), absence of such a pole requires that ). While, as discussed in section uv q c +1 + 3.19 , ), all vertex operators of type + 3.14 2 3.18 N 0 = / = 0 states at the expense of introducing further fermion field bc 2( c in the BRST operator with the corresponding ghost field 4.21 +3 , = U(1) 2 c ), was the truncation to the zero-charge states with respect q / U(1) and the stress tensor. J 2.6 +1 F = 6 super-Chern-Simons theories no two adjacent external st , J 2 / ) breaks up into a sum over instanton sectors, with the zero-i N 1 defined in eq. ( = 6 conformal supergravity proposed in [ − 3.9 A feature of the construction discussed in the previous sect An important step in our construction, which led to a spectru Gauging this symmetry requires that the fermion action be co ( These charges and multiplicities ensure that there is no ano N 21 = ( U(1) ij example, with four fermions (Ψ q truncation to structures of eq. ( stripped amplitude belong toratios the of same the multiplet. rank For of produc the factors it is possible to change this points. This was, of course, a consequence of their It may be possible to accommodate such values of in the similar localization does seem tonation occur for for the the color-fermi absenceto of explore the whether duality the for duality ABJ(M)same is amplitudes degree restored if as [ one the restricts “kinematic t instantons”. producing the amplitudes ( function and thus theadditional current terms correlators. can be Itframework, in given would which both an be the interpretation inter kinematicrically and from color (and a part are of field an given am The t by instanton kinematic sums), part is however reminiscent exhibits of localization on instan this end there should be no second order pole in the multiplets ( at the classical level, quantum mechanically it requires th inclusion of the current central charge nilpotent. Using eq. ( to with the gauge-nonsinglet fields is given by the Lagrangian o Thus, with the choice ( J to the new gaugefor topologically-nontrivial field. sectors. Thus, As theJ with correlat the GL(1) gauge field, it may be ga JHEP06(2014)088 ) 2 2) N − ) we , the (6.4) (6.3) to the ij 6.3 J i 1 ) 4 − ρ U ( ) factors and F 32 ]. These poles , P N 0 ). In the latter 39 V ). This example = ) 3 i ) ρ 6.3 4.21 ,ij 6 ( 1 ρ − 43 ( , V 1 32 , − r the ABJ(M) theory, the 0 V V ors, respectively. Similarly ) nder different gauge group ) and ex operators are of type Ψ 2 nder the same gauge group s a standard quantum field 5 ucture for the corresponding non-local quartic Lagrangian uge field (if the uncontracted n of eq. ( ρ f a similar type), identify the ρ est that it is either due to the roperties of these and higher- f freedom (if the uncontracted ij ( ed to eq. ( ) and three SU( ( oint amplitudes following from J -channel uncontracted fermions For the first amplitude ( 14 ) is a function of dimension ( quires that two multiplets of the 0 , 43 N t nce of such terms in the effective field i , 1 1 U he same footing as regular Chern-Simons 23 − ij − F h V V ( tructed in our formalism. ) P ) ) (6.5) 1 f 4 i - and ρ = ρ s ( ij ( h 21 ( ,ij , 14 , 0 0 f 1 ) V -channel factorization poles [ V − t h Q V ), the ( ) – 22 – 6 and thus both transform under the first SU(5 3 δ 6.3 ρ and 1 and ( = i ) are the same as the poles of the four-point ABJ(M) − 21 ) , ) while the uncontracted fermions between the second s 4 4 0 ρ A 6.3 N V ( ) 2 12 , ρ 1 ( − 12 , V 1 ) 3 − ρ V -channel poles of this amplitude may potentially be related ( ) t 1 21 , ρ 0 ( V 21 ) , 1). Defining the operators 2 - and 0 ρ s − V ( , h 12 , +1 1 , − or from a non-local contact interaction term. V ) are on-shell supersymmetry generators and +1 1 , 22 ρ Q ( +1 Since the kinematic part of vertex operators is the same as fo 21 , , It appears difficult to rule out on general grounds the appeara While this may appear strange at first sight, such fields are on t = 6 superconformal symmetry is 0 1 23 22 V − h we see that, for the second amplitude ( exchange of Chern-Simons field for these two gauge group fact appears to also beamplitude nonvanishing and to has factorize the into correct the color product str of amplitudes relat states they factorize ontheory with and this understand tree-level S-matrix. whether there exist transform under different gauge-group factorsexchange and of a thus bi-fundamental sugg unphysical fieldterm. or it It is would due be topoint a interesting amplitudes to (as explore well as the in factorization more p general constructions o matter in bifundamental representation. theory Lagrangian. see that the uncontractedand fermions thus between both the transform under first SU( two vert may arise either from thefermions exchange of between a two Chern-Simons-like adjacent ga factor), vertex a bi-fundamental operators field transform withfermions u no between asymptotic degrees two o adjacentfactors) vertex operators transform u N appears to describe a product gauge group with one SU(5 amplitudes. The general structure of color-ordered four-p same type are adjacent. Moreover, the correlator the current correlator has a single trace structure which re gauge fields, for which no physical vertex operator can be cons and third vertex operators are of type Ψ ( correlators which thus necessarily has nontrivial are nonvanishing. The former leads to the four-point versio where factor; thus, the poles of the four-point amplitudes ( JHEP06(2014)088 1 , − For , , , s (of 24 6= 0 on ommons k ]. charge. with k U U(1) SPIRE rgy under contract F , q -Mills Amplitudes d therefore the field IN e aft. OTE would like [ P n for useful discussions ]. ossible to include states 2 as well as the potential ]. ishing 31 redited. tors of – ≤ − amental representation. The ty while this paper was being SPIRE ]. ]. n 29 IN nstructed following the analysis can be used to construct currents ][ 2 q equires that all states are kept and perhaps SPIRE SPIRE ). For suitable choices the relation ± 2 IN IN arXiv:1308.1697 q rspective. ]. , ][ ][ − and 1 1 q factors with q ( SPIRE n ± ± IN On the tree level S matrix of Yang-Mills theory U [ – 23 – hep-th/0312171 [ ]. = 6 superconformal symmetry; if so, it would be ) and Scattering of Massless Particles: Scalars, Gluons and Scattering of Massless Particles in Arbitrary Dimension 2 N q hep-th/0612250 hep-th/0403190 [ [ SPIRE 1 suggest that such amplitudes are very constrained. + IN (1989) 616 ), which permits any use, distribution and reproduction in [ ≥ 1 Scattering amplitudes (2004) 189 ]. q n ( ± Multi-Gluon Cross-sections and Five Jet Production at Hadr 252 , B 312 2 SPIRE q IN 2 [ it may be possible to construct a theory with four gauge group CC-BY 4.0 (2007) 025028 (2004) 026009 ± , F This article is distributed under the terms of the Creative C 1 q factors with q 2 D 75 Perturbative gauge theory as a string theory in twistor spac D 70 n arXiv:1309.0885 Nucl. Phys. On the Factorisation of the Connected Prescription for Yang ± , U and , 2 , 1 q Colliders arXiv:1307.2199 Phys. Rev. Commun. Math. Phys. Phys. Rev. Similarly, by including fermions of different charges it is p In both constructions above it appears that the S-matrix — an One may also relax this constraint; in this case consistency r 24 [1] H. Elvang and Y.-t. Huang, [7] F. Cachazo, S. He and E.Y. Yuan, [4] R. Kleiss and H. Kuijf, [5] C. Vergu, [6] F. Cachazo, S. He and E.Y. Yuan, [2] E. Witten, [3] R. Roiban, M. Spradlin and A. Volovich, unrelated ranks) and eighttree-level matter scattering multiplets amplitudes in ofin such the this theories bi-fund paper; can the be high co degree poles in the between with charges any medium, provided the original author(s) and source areReferences c and Y.-t Huang, N.to Obers thank the and Niels A. Bohrwritten Tseytlin International up. for Academy comments for hospitali onDE-SC0008745. This the work dr is supported byOpen the Access. US Department of Ene Acknowledgments We would like to thank Y.-t Huang, H. Johansson and A. Tseytli created by a finite number of vertex operatorsexample, containing two fac pairs of fermions with charges zeroes in the while continuing to truncate the spectrum to states with van the theory is better thought of from a higher-dimensional pe Attribution License ( interesting to understand how it evades the arguments of [ theory generating it — may have JHEP06(2014)088 , , , , , tudes in , ]. (2008) 091 (2009) 016 SPIRE , ]. , IN 10 01 Chern-Simons ][ ]. SPIRE super Yang-Mills = 6 , ]. IN The Four-Loop Planar (2010) 045016 c Chern-Simons Theory JHEP JHEP N superconformal ][ , = 4 ymmetric Yang-Mills ]. (2011) 116 , Dual superconformal SPIRE ]. N SPIRE v, 03 rnov, IN = 6 D 82 Magic identities for conformal IN ]. ][ SPIRE A duality for the S matrix N Unification of Residues and y duals ][ ]. Tree-level Recursion Relation and IN SPIRE arXiv:1207.4851 ][ JHEP IN [ SPIRE , ][ IN SPIRE ]. ]. ]. 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