arXiv:1507.02936v1 [hep-th] 10 Jul 2015 l aesonta n a o decouple not can at one Green that hand, shown other have the On a al. provide re- point. could the starting that promising 10d hope for might developments One cent work from. our motivation context 4 the its recast is draws This to possible manifest. were become might it ness If [7]. the- not this or N whether finite question UV open is important ory an is It sions. inw ilstu h eea rmwr.Pr fthis of Part framework. general the up set will we central tion reproducing time, same supergravity. -like the of possesses properties at amplitudes while, 1-loop we features un- the prescription an for natural contains it propose the because symmetry, theory Virasoro naively stringy gauged not a the does be though supergravity to Even appear 4d for amplitudes. model loop ambitwistor investigate to is tree- goal correct the of reproduce amplitudes to level David shown by was developed that [9] model Skinner a of version ambitwistorial an superstring 10d of [8]. theory compactification the from pergravity vr is ever, a the- from string amplitudes [3–6]. ambitwistor ory scattering as level known calcula- loop description the worldsheet and for tree allowed of theories, has tion supergravity 10d progress For recent source results. were valuable remarkable worldsheet and a they ideas became these theory new still field for of they the describe, most to to though equivalent designed string Even not twistor are of models 2]. context [1, the in models primarily theory, and field string between dualities to coupling attempts weak-weak of construct number with a motivated strong/weak This relation by duality. obscured coupling somewhat precise is a cor- theory the particle find AdS/CFT the formulation, to the framework theoretic While systematic string a provides . respondence particle conventional into of insights models unexpected and deep uncover h lno hsnt sa olw.I h etsec- next the In follows. as is note this of plan The using by here route different a pursue therefore We n ftems neetn rvttoa hois how- theories, gravitational interesting most the of One can theories field quantum of descriptions Worldsheet uegaiyi h omo tigter,finite- theory, string a of form the in supergravity 8 = N 1 uegaiyin supergravity 8 = I nt ¨rTertsh hsk nvriyo Hamburg, of University Physik, f¨ur Theoretische Inst. II. uha xiiigmdlrivrac n erdcn h e the that reproducing amplitudes and invariance genus-one amplitudes. modular for supergravity exhibiting definition leve as a tree such supergravity proposing to by algebr rise work the give current correlators to appropriate worldsheet correspond an operators gauging vertex After ambitwistors. nti oew drs h olsetdsrpino 4-dime of description worldsheet the address we note this In .INTRODUCTION 1. 2 EYTer ru,DS abr,Ntetas 5 -20 H D-22607 85, Notkestrasse Hamburg, DESY Group, Theory DESY N oad olsetDsrpinof Description Worldsheet a Towards uegaiyi 1] u main Our [10]. in supergravity 8 = d pc-iedimen- space-time 4 = rhrLipstein Arthur N N su- 8 = Dtd uy2015) July (Dated: uemlilt thspeiul ensonthat shown been previously has It supermultiplet. 8 = 1 d n okrSchomerus Volker and n eaiehlct ttso the of positive states sector, to helicity Ramond negative correspond a and operators of vertex and physical inclusion operator only the vertex the after physical even a that, of we argue notion addition In the define 10]. [9, shall from constructions reviews simply esi h atrsco.T emr rcs,ltus let precise, more fields be To bosonic free sector. the matter introduce the in tems erdcsteepce Rlmt.W ocuewt a with problems. open conclude We and results limits. of in- IR discussion modular expected is the and it conservation that reproduces in show momentum we computed 4 satisfies and section variant, defined in and then 3, is section amplitude 1-loop The h cino h atrsco ae h form the takes sector matter respectively. the sector, ghost of and action matter The the as to refer shall ∂φ the iwso oe,w utbsnz oeo the of some bosonize must we model, bitwistor n ˜ and n ih r ooi.Ltu oeta h oa central is total multiplets the of that sets note two us these Let of charge remain- bosonic. the are while eight fermionic ing are components four first the where oss ofra weight conformal possess iey h eodpi fmultiplets of pair second The tively. W ytm fcnomlweight conformal of systems ue,seeg 1] hi xoetasexp( exponentials their [11], e.g. see rules, ih r emoi.Te omfour form remaining They the while fermionic. bosonic are are eight components four first h miwso oe ossso w etr htwe that sectors two of consists model ambitwistor The nodrt pl u h etxoeaoso h am- the of operators vertex the out spell to order In λ (˜ = Z W ρ ˜ = cteigapiue.W xedthis extend We amplitudes. scattering l uue huse19 -26 Hamburg, D-22761 149, Chaussee Luruper µ Z (˜ = µ α .TEABTITRTHEORY AMBITWISTOR THE 2. sse xetta l rdnsaervre,i.e. reversed, are gradings all that except -system α and , ,w ru htteol physical only the that argue we a, λ λ pce nrrdbhvo f1-loop of behavior infrared xpected ˜ N ρ α α nsional ˙ α asssvrlcnitnychecks consistency several passes ˜ ; S , and W χ 8 = ρ m ˜ a ..Abtitrfields Ambitwistor 2.1. α ˙ with ) ∼ ˜ ; eoemultiplets denote ω 2 ∂φ N Z a Supergravity ntemte etri iia to similar is sector matter the in ) Σ µ uegaiyusing supergravity 8 = d mug Germany amburg, = α, 2 z µ α h ˙ W α ~ s ˙ 1 = λ ˜ = α · ˙ h olwn h standard the Following . ∂Z Z ¯ , − and 2 s 1 = N φ λ 2 ˜ + Z λ − supermultiplet. 8 = ρ ( = / and s · a = 2 µ 2 βγ ρ ∂ρ ¯ c esalcon- shall We . 1 = λ 0. = ( = α  s φ n eight and µ , λ µ h , . . . , φ W ρ α ˙ λ uhthat such α ; respec- , + ρ , χ βγ .The 8. a α s ˙ and ) µ ; sys- ω φ µ a bc ) ) 2 sider operators of the form Now we turn to the ghost sector, which contains two multiplets C = (C ) = (c ,c ,c ,c ; γ ,γ , γ¯ , γ¯ ) s φ +s φ A g h + − + − + − Φ= ϕ(W, Z, ρ, ρ˜)e λ λ µ µ , and B = (BA) accordingly. Note that the index A runs over our basis of currents. All eight components of the where ϕ is any expression composed from components of multiplet C have conformal weight h = 0 while those the arguments and derivatives thereof, and refer to these C in B possess h = 1. Following the usual bosonization as operators in the (s ,s ) sector of the theory. One B λ µ of ghost systems, we shall introduce two bosonic fields φ family that will play an important role below is given by + − and φ such that ∂φ = β γ+ + β γ− and similarly for dt α a ∂φ. An operator is said to be in the (p, p¯) picture if it ˜(λ,λ˜; η) := eit(˜µ (z)λα+χ ˜a(z)η )δ2(˜λ tλ˜(z)) (1) H t3 − contains the factor Z pφ+¯pφ which are parametrized by the bosonic variables λ,λ˜ e with h~p = p(p + 1) p¯(¯p + 1) . along with the fermionic variables η. In order to clearly − − distinguish the parameters from the fields, we have dis- From the fields in the ghost sector we can build a played the dependence on the worldsheet coordinate z in GL(1 1) ⋉ R2|2 current algebra at level k = 2 using this formula. Using the standard identification δ2(˜λ) the standard| prescription and ultimately the usual− BRST ∼ exp( φ ) we conclude that the fields ˜ belong to the current Q. For the theory to be anomaly free and nilpo- − µ H (s ,s ) = (0, 1) sector of the theory. It is not diffi- tency of the zero mode Q0, it is crucial to have = 8 λ µ N cult to check that− the integral over t projects onto the . component of conformal weight is hH˜ = 1. There has been some discussion whether the c = 0 Using the state-field correspondence, the ground state stress tensor of the ambitwistor theory is gauged or not. 0, 1 exp( φµ(0)) vac of the (0, 1) sector satisfies We have verified that the combined stress tensor of the the| − followingi∼ modified− | vacuumi conditions:− ghost and matter sectors is non-trivial in the cohomology of the current algebra BRST operator Q0. One way out µα˙ 0, 1 =0λ ˜ 0, 1 = 0 (2) n+1| − i α,n˙ −1| − i could be to add further ghosts for the of the model. But this would require to add additional mat- Z 1 Z for 0 < n + 2 , and is annihilated by Kn, 0 < n ter fields with c = 26, which appear rather artificial. We for all other∈ matter fields. This means that the modes∈ α˙ believe that the Virasoro symmetry of the ambitwistor µ1/2 are creation operators even though they lower the model should be considered accidental, just as the trans- conformal weight, whileλ ˜α,˙ −1/2 are annihilation opera- verse Virasoro algebra that appears for flat space string tors in the (0, 1) sector. Such a partial swap of creation − theory in light-cone gauge. Consequently, we suggest not and annihilation operators is a crucial ingredient in the to gauge the stress tensor of the ambitwistor model. oscillator construction of particle multiplets, see e.g. [12] for the case of = 8 supergravity. It is the key mecha- N nism by which the 4d ambitwistor model accommodates 2.3. Physical vertex operators the = 8 supergravity multiplet. N We can now specify what we mean by physical vertex 2.2. The BRST operator operators of the ambitwistor model. By definition, these are scale-invariant integrated vertex operators in the co- The 4d model proposed by David Skinner in [9] gauges homology of this BRST operator Q0. The integration is a current algebra consisting of four bosonic currents over the chiral worldsheet coordinate z so that the field in the integrand must have conformal weight h = 1. g = : Z W : , h =: ρ ρ˜ : (3) It is not too difficult to check that the following vertex · · operators in the ( 1, 0) picture are physical: e+ = ρ,ρ , e− = [˜ρ, ρ˜] (4) − h i ˜(−1,0) ˜ and four fermionic currents (λ,λ˜; η)= dzδ(γ+)δ(γ−) (λ,λ˜; η) . (7) V H Z f + = Z,ρ , f − = Z ρ˜ (5) h i · We shall refer to the integrand of this vertex operator ¯+ ¯− as ˜(−1,0)(λ,λ˜; η)(z). In order to read off the picture, f = ρ W, f = [W, ρ˜] . (6) V · we recall that the ghost operator δ(γ+)δ(γ−) exp( φ). The brackets , and [, ] instruct us to project to the The physical vertex operators (7) turns out∼ to describe− components carryingh i an undotted index α and a dotted negative helicity states. Similarly, one can also construct indexα ˙ , respectively, and to contract the indices with an physical vertex operators for positive helicity states. In ǫ tensor. It is easy to check that these currents form a the (0, 1) picture they takeV a very similar form and are GL(1 1) ⋉R2|2 current algebra at level k = 2. The latter essentially− obtained by complex conjugation. After per- may be| obtained as a contraction of an SL(1 2) current forming complex conjugation, the vertex operators will algebra. | depend on the fermionic variablesη ˜. For computational 3 purposes however, it is convenient to Fourier transform We have spelled out vertex operators in the pictures η˜ into η so that, see [10] for more details, (0, 1), ( 1, 0) and (0, 0). On the sphere we can insert one− operator− in the (0, 1) and ( 1, 0) picture each and − − (0,−1)(λ,λ˜; η)= dzδ(¯γ )δ(¯γ ) (λ,λ˜; η), then put all others in the (0, 0) picture. For the torus V + − H Z all vertex operators may be put in the (0, 0) picture. In where the following we will simply refer to such vertex opera- tors as and ˜. Tree-level amplitudes were computed dt α V V (λ,λ˜; η) := eitµ (z)˜λα δ2|8 (λ tλ(z) η tχ(z)) . in [10] and they were shown to coincide with the n-point H t3 − | − Nk−2MHV amplitudes of = 8 supergravity in four di- Z N To obtain the physical vertex operators of positive and mensions. Let us point out that, contrary to the impres- negative helicity states in other pictures, we can apply sion that is given in [10], the computations do not involve the following picture-changing operators a ghost sector for the Virasoro algebra of the ambitwistor theory. X = δ(β+)δ(β−)f +f − , X = δ(β¯+)δ(β¯−)f¯+f¯− . (8) Our main goal in this section is to propose an expres- sion for the 1-loop amplitude of the ambitwistor theory. In particular, the integrand of a physical vertex operator To this end we take the (Z, W ) to possess periodic bound- for negative helicity states in the (0, 0) picture is given ary conditions, while (ρ, ρ˜) can be either periodic or an- by tiperiodic. The 1-loop amplitudes will then involve a sum (0,0) (−1,0) over spin structures as well as a GSO projection. For even ˜ (λ,λ˜; η)(z)= X(w)˜ (λ,λ˜; η)(z) k−2 V V w→z spin structure, the 1-loop n-point N MHV amplitude   is given by ∂ ˜ ∂ ∂ ˜ = Z, H + ρ, ρ˜ H (λ,λ˜; η)(z). "* ∂W + ∂W · ∂W # (1) = dτΠn dz Πk ˜ Πn ,   Aeven i=1 i l=1Vl r=k+1Vr Similarly, application of the picture changing operator Z D E where τ is the complex structure of the torus, and there is X to (0,−1) gives the vertex operator (0,0) for positive an implicit integral over the zero modes of the b,c ghost helicityV states in the (0, 0) picture. V zero modes. Combining the exponentials of the vertex In order to argue that the ambitwistor model contains operators with the action and integrating out (µ, µ,˜ χ˜) no further physical vertex operators for particle multi- implies the equations of motion plets, we can employ the relation with oscillator represen- tations (see comment above) to show that the ~s = (0, 1) k − ¯ and (1, 0) sectors are the only ones that can accommo- ∂λ = tlλlδ (z zl) − date infinite multiplets including arbitrary derivatives of Xl=1 the space time fields. In a second step one determines the n ∂¯λ˜ = t λ˜ δ (z z ) cohomology of the BRST operator Q0 in the L0 = 1 sub- r r − r space of these two sectors. Below we shall suggest that r=Xk+1 the ambitwistor theory should also include a Ramond sec- k ¯ tor in which theρρ ˜ system possesses anti-periodic bound- ∂χ = tlηlδ (z zl) . (9) − ary conditions. We claim that such a Ramond sector does Xl=1 not give rise to additional physical vertex operators ei- For a genus-one worldsheet, the solutions are ther. In this sense, the ambitwistor theory contains no more than the = 8 supergravity multiplet. A more k N λ(z) = λ + t λ S (z z , τ) (10) detailed derivation of these statements will be given in a 0 l l 1 − l forthcoming paper. l=1 Xn λ˜(z) =λ ˜ + t λ˜ S (z z , τ) (11) 0 r r 1 − r 3. SCATTERING AMPLITUDES r=Xk+1 k χ(z) = η + t λ S (z z , τ) (12) Let us now turn to a discussion of scattering ampli- 0 l l 1 − l tudes. Since we have not gauged the Virasoro symmetry Xl=1 of the model, we cannot employ the usual formulae for where S1(z, τ) is the Green’s function on a torus with amplitudes which involve Virasoro b-ghosts and Beltrami periodic boundary conditions, and (λ0,λ˜0, η0) are zero differentials. The prescriptions we shall discuss here look modes. Note that there appear three other types of very natural, though they are justified only a posteriori Green’s functions, which correspond to having antiperi- when we establish the relation with supergravity ampli- odic boundary conditions along at least one direction of tudes. the torus. We denote them by S2,3,4(z, τ). When these In order to have non-vanishing amplitudes on a surface solutions are plugged into the delta functions of the ver- of genus g, the total picture charges of all inserted ver- tex operators, this gives rise to the 4d genus g = 1 scat- tex operators should add up to (p, p¯) = (g 1,g 1). tering equations refined by helicity. Let us note that − − 4 solutions to the equations of motion can only exist if the are invariant under τ τ +1. As a result, we integrate sum of residues vanishes, τ over the fundamental→ domain of the complex plane. Next we verify that the amplitude encodes momentum Rλ = Rλ˜ = Rχ =0 , conservation. To see this, consider the sum of the mo- menta of the positive helicity particles where n n k k n k λrλ˜r = trλ˜r λ0 + tlλlS1 (zrl, τ) . Rλ = tlλl, Rλ˜ = trλ˜r, Rχ = tlηl. ! r=Xk+1 r=Xk+1 Xl=1 Xl=1 r=Xk+1 Xl=1 To obtain the first equality, we used the delta functions The remaining contributions to the correlation function in eq. (14) to replaceλ ˜r with trλ˜ (zr) and then employed arise from integrating over fluctuations, which gives a 1- eq. (11). Noting that loop partition function, and evaluating contractions of the ρ, ρ˜ fields, which gives a determinant. The one-loop n k k n partition function can be composed from the partition t λ˜ t λ S (z )= t λ t λ˜ S˜ (z ) r r l l 1 rl − l l r r lr function of the components through standard formulas. r=k+1 l=1 l=1 r=k+1 In the end, one obtains X X X X and inserting eqs. (10) and (15), one finds that 2 2 8 (1) d λ0 d λ˜0 d η0 n dzi dti n n k even = dτ Πi=1 3 A GL(1) ti Z λiλ˜i = λ0 trλ˜r +˜λ0 tlλl. 2 2 8 k n δ (Rλ) δ (Rλ˜) δ (Rχ) Π δl Π δr M (13) i=1 l=1 r=k+1 X r=Xk+1 Xl=1 where   The right-hand-side vanishes on the support of the delta functions δ (Rλ) and δ (Rλ˜), so momentum is conserved. 2 δl = δ (˜λl tlλ˜ (zl)) (14) Similar considerations apply for supermomentum conser- − 2|8 vation. δr = δ (λr trλ (zr) ηr trχ (zr)) (15) − | − −4 In summary, the bosonic delta functions in eq. (13) M = ( 1)α det H θ (0, τ)/η(τ)3 . (16) − α α provide 2n constraints on the integration variables, since α=2,3,4 X  four of the delta functions ultimately encode momentum conservation. On the other hand, there are 2n+4 bosonic Here H is an n n block-diagonal matrix whose nonzero α × integrals, so in the end there are four remaining integrals elements are given by which can be interpreted as an integral over loop momen- tum. In particular, after the delta functions localize the Hlm = t t λ , λ S (z , τ) , l = m α l m h l mi α lm 6 2n coordinates of the vertex operators, one is left with an Hll = t λ (z ) , λ (17) α l h l li integral over the zero modes (λ0,λ˜0), and τ. If we make the change of variables τ = log q, the resulting integral for l,m 1, ..., k , and ∈{ } is similar to one that would arise from on-shell diagrams [13], where q would be interpreted as a BCFW shift. In Hrs = t t [˜λ ,λ˜ ] S (z , τ) , r = s α r s r s α rs the present context, q is related to the complex structure rr 6 Hα = tr [˜λ (zr) ,λ˜r] (18) of the worldsheet, and the integral over q contains a UV cutoff due to modular invariance. for r, s k +1, ..., n . The 1-loop amplitude for odd ∈ { } As a final consistency check, we verify that the am- spin structure can be computed in a similar manner. We plitudes exhibit the expected IR divergences, using tech- will not need the form of these contributions and there- niques developed in [5, 14, 15]. In the IR limit, τ fore refrain from giving any details here. and the worldsheet degenerates to a sphere withℑ two→ ad- ∞ ditional punctures which are associated with vertex oper- ators with equal and opposite momentum k = λ0λ˜0. We 4. CONSISTENCY CHECKS will denote the additional vertex operators as ˜a, b. In this limit, the contribution from odd spin structureV V van- In this section, we will describe various consistency ishes, and the sum of determinants in eq. (16) simplifies checks for the amplitudes computed in the previous sec- substantially. Noting that η(τ) q1/24 and tion. For concreteness, we will focus on the formula for → 1-loop amplitudes with even spin structure in eq. (13). q1/8, α =2 θ (0, τ) First note that the amplitude is modular invariant, which α → 1, α =3, 4 is suggestive of an interpretation as a am-  plitude. This can be seen by noting that if τ 1/τ, where q = e2iπτ , one finds that then z z/τ, t t/√τ, (Z, W ) √t(Z,→ W ), − and θ /η3 →τ −1 θ /η→3 . Furthermore,→ all of these terms M det H + q (det H det H ) . α → α → 2 4 − 3  5

Furthermore, since We have also proposed a new formula describing 1- loop amplitudes of = 8 supergravity which appears 1/z, α =2 N S (z, τ) to have a very different structure than the 1-loop am- α → constant, α =3, 4 plitude of 10d ambitwistor string theory. For example,  in the 10d formula, there is an additional delta function µ we see that the contributions from α =3, 4 cancel out and which enforces that PµP = 0 vanish at some point on M det H , which is the same determinant appearing in µ → 2 the worldsheet, where P is the canoncial momentum of tree-level supergravity amplitudes [10]. Hence, in the IR the worldsheet theory. This constraint ensures that only limit a genus-one amplitude reduces to an (n + 2)-point massless states can appear in factorization channels, but tree level amplitude integrated over the on-shell loop mo- there is no such delta function in the 4d formula since mentum k. Moreover, the IR divergent part of the ampli- this constraint is already built into the formalism. Fur- tude comes from the region of integration k 0. In this thermore, whereas in the 10d formula the delta functions limit, we can Taylor expand ˜ , in the soft→ momentum Va Vb determine the modular parameter τ, in the 4d formula and keep the leading order terms. The IR divergent part the delta functions leave τ unfixed giving rise to a 4d loop of the loop integrand can then be determined by using integral which is reminiscent of the formulae one obtains Stokes theorem to express the soft vertex operators as for 1-loop amplitudes of = 4 super-Yang-Mills the- contour integrals, integrating them around each pair of ory using on-shell diagrams.N It is intriguing to see such hard vertex operators, and adding up the residues. In structure emerging in the context of 4d gravitational am- the end, we obtain plitudes. (ǫ k )2 (ǫ∗ k )2 (1) = d4kδ k2 · i · j (0), When computed on the torus, the loop amplitudes of An |div k k k k An i,j i j ambitwistor models depend on elliptic functions and their Z X · ·  equivalence to field theory amplitudes is difficult to see. αβ˙ α β˙ In a recent paper [6], this difficulty was overcome for 10d where ǫ = ξ λ˜0 / ξλ0 and ξ is a reference spinor. Evaluating the this integralh i using dimensional regular- model by transforming the 1-loop amplitude to a sphere, ization then gives the standard 1-loop IR divergences of and it would be very interesting to implement a similar = 8 supergravity [16]. mapping of our 4d 1-loop amplitude. Although the loop N integrands of the 4d ambitwistor model contain elliptic functions, they may still be useful for studying various 5. CONCLUSION AND OUTLOOK properties of = 8 supergravity. For example, since a genus g amplitudeN of the 4d ambitwistor string model should encode all g-loop Feynman diagrams of = 8 su- Above we argued that the spectrum of the critical, N non-anomalous 4d ambitwistor model corresponds pre- pergravity with a given number of external legs, it should cisely to = 8 supergravity, and define a prescription exhibit better power counting behavior than individual for computingN loop amplitudes in this model. In doing Feynman diagrams. Ultimately, we hope that the 4d am- bitwistor model reveals new symmetries of = 8 super- so, it is not necessary to gauge the Virasoro symmetry N as one does for the 10d ambitwistor string or the stan- gravity which provide insight into its possible finiteness. dard RNS string, since scale invariance and current al- gebra symmetry of the world sheet theory appear to be Acknowledgements: We thank Rutger Boels, Lionel powerful enough to remove all unphysical states. Hence, Mason, and especially David Skinner for helpful discus- the Virasoro symmetry can be thought of as an acciden- sions. This work was supported by the German Sci- tal symmetry which does not impose further constraints ence Foundation (DFG) within the Collaborative Re- on the spectrum. Recent studies of gravitational soft search Center 676 Particles, Strings and the Early Uni- theorems using 4d ambitwistor model also suggest that verse and by the People Programme (Marie Curie Ac- Virasoro symmetry does not play an essential role as a tions) of the European Union’s Seventh Framework Pro- symmetry of the gravitational S-matrix [15], so it would gramme FP7/2007-2013/ under REA Grant Agreement be interesting to make this connection more concrete. No 317089 (GATIS).

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