Towards a Worldsheet Description of =8 Supergravity N Arthur Lipstein1 and Volker Schomerus2 1 II. Inst. f¨ur Theoretische Physik, University of Hamburg, Luruper Chaussee 149, D-22761 Hamburg, 2 DESY Theory Group, DESY Hamburg, Notkestrasse 85, D-22607 Hamburg, Germany (Dated: July 2015) In this note we address the worldsheet description of 4-dimensional N = 8 supergravity using ambitwistors. After gauging an appropriate current algebra, we argue that the only physical vertex operators correspond to the N = 8 supermultiplet. It has previously been shown that worldsheet correlators give rise to supergravity tree level scattering amplitudes. We extend this work by proposing a definition for genus-one amplitudes that passes several consistency checks such as exhibiting modular invariance and reproducing the expected infrared behavior of 1-loop supergravity amplitudes. 1. INTRODUCTION Worldsheet descriptions of quantum field theories can simply reviews constructions from [9, 10]. In addition we uncover deep and unexpected insights into conventional shall define the notion of a physical vertex operator and models of particle physics. While the AdS/CFT cor- argue that, even after the inclusion of a Ramond sector, respondence provides a systematic framework to find a the only physical vertex operators correspond to positive and negative helicity states of the = 8 supermultiplet. string theoretic formulation, the precise relation with N the particle theory is somewhat obscured by strong/weak The 1-loop amplitude is then defined and computed in coupling duality. This motivated a number of attempts to section 3, and in section 4 we show that it is modular in- construct weak-weak coupling dualities between field and variant, satisfies spacetime momentum conservation and string theory, primarily in the context of twistor string reproduces the expected IR limits. We conclude with a models [1, 2]. Even though most of these worldsheet discussion of results and open problems. models are not equivalent to the field theory they were designed to describe, they still became a valuable source for new ideas and results. For 10d supergravity theories, 2. THE AMBITWISTOR THEORY remarkable recent progress has allowed for the calcula- tion of tree and loop level scattering amplitudes from a 2.1. Ambitwistor fields worldsheet description known as ambitwistor string the- ory [3–6]. One of the most interesting gravitational theories, how- The ambitwistor model consists of two sectors that we ever, is = 8 supergravity in d = 4 space-time dimen- shall refer to as the matter and ghost sector, respectively. sions. ItN is an important open question whether this the- The action of the matter sector takes the form ory is UV finite or not [7]. If it were possible to recast 4d = 8 supergravity in the form of a string theory, finite- S d2z W ∂Z¯ +ρ ˜ ∂ρ¯ N m ∼ · · ness might become manifest. This is the context our work ZΣ draws its motivation from. One might hope that the re- α˙ a cent developments for 10d supergravity could provide a where Z and W denote multiplets Z = (λα,µ ; χ ) and α promising starting point. On the other hand, Green at W = (˜µ ,λ˜α˙ ;˜χa) with α, α˙ =1, 2 and a =1,..., 8. The arXiv:1507.02936v1 [hep-th] 10 Jul 2015 al. have shown that one can not decouple = 8 su- first four components are bosonic while the remaining pergravity from the compactification of 10dN superstring eight are fermionic. They form four βγ and eight bc theory [8]. systems of conformal weight hZ = 1/2 = hW , respec- α˙ a We therefore pursue a different route here by using tively. The second pair of multiplets ρ = (ρα,ρ ; ω ) α an ambitwistorial version of a model developed by David andρ ˜ = (˜ρ , ρ˜α˙ ;˜ωa) in the matter sector is similar to Skinner [9] that was shown to reproduce the correct tree- the WZ-system except that all gradings are reversed, i.e. level amplitudes of = 8 supergravity in [10]. Our main the first four components are fermionic while the remain- goal is to investigateN loop amplitudes. Even though the ing eight are bosonic. Let us note that the total central ambitwistor model for 4d supergravity does not naively charge of these two sets of multiplets is c = 0. appear to be a stringy theory because it contains an un- In order to spell out the vertex operators of the am- gauged Virasoro symmetry, the natural prescription we bitwistor model, we must bosonize some of the βγ sys- propose for the 1-loop amplitudes possesses string-like tems in the matter sector. To be more precise, let us features while, at the same time, reproducing central introduce the free bosonic fields φλ and φµ such that α α˙ properties of supergravity. ∂φλ =µ ˜ λα and ∂φµ = µ λ˜α˙ . Following the standard The plan of this note is as follows. In the next sec- rules, see e.g. [11], their exponentials exp(sλφλ + sµφµ) tion we will set up the general framework. Part of this possess conformal weight h = s2 s2 . We shall con- ~s − λ − µ 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 supersymmetry. 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 Virasoro algebra 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).