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A worldsheet theory for

Citation for published version: Adamo, T, Casali, E & Skinner, D 2015, 'A worldsheet theory for supergravity', Journal of High Energy , vol. 2015, no. 2, 116. https://doi.org/10.1007/JHEP02(2015)116

Digital Object Identifier (DOI): 10.1007/JHEP02(2015)116

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Download date: 05. Oct. 2021 JHEP02(2015)116 Springer tent at the January 5, 2015 October 16, 2014 January 25, 2015 : ure terms. The February 18, 2015 ved target space : lies in Field and : : , Revised t space fields obey the 10.1007/JHEP02(2015)116 Received Accepted Published as quantum corrections to doi: scattering equations to curved niversity of Cambridge, Published for SISSA by [email protected] , . 3 -field and . The conditions for the theory to be consis 1409.5656 B The Authors. c Classical Theories of Gravity, Supergravity Models, Anoma

We present a worldsheet theory that describes maps into a cur = 10 supergravity equations of motion, with no higher curvat , [email protected] d [email protected] Department of Applied MathematicsWilberforce & Road, Theoretical Cambridge Physics, CB3 U 0WA, UnitedE-mail: Kingdom these curved space scattering equations. Keywords: path integral is constrained tospace. obey a Remarkably, generalization the of supergravity the field equations emerge quantum level can benonlinear computed exactly, and are that the targe ArXiv ePrint: Theories equipped with a Open Access Article funded by SCOAP Abstract: Tim Adamo, Eduardo Casali and David Skinner A worldsheet theory for supergravity JHEP02(2015)116 7 9 1 3 5 6 10 12 ometry, or ]. -functionals, 7 β – ]. This infinite 4 10 , 9 ourse different aspects and Φ [ expansion of amplitudes ]. The relationship can ′ 3 B α – 1 ively difficult to write down motion that at low energies works perturbatively in the orldsheet on an for r for a vertex operator to be finitesimal fluctuations of the the worldsheet ht, reflecting the fact that the from the requirement that the Relativity emerges as the low on-linear sigma model. Higher orldsheet Weyl anomaly. orldsheet CFT are infinitesimal f the rst observed via the tree-level S- ity [ t space field equations, linearized n guaranteeing the excellent high -field, and dilaton Φ. Maintaining B , g – 1 – -function of the superstring [ 0 limit of a sphere amplitude in gives β → ′ α , which governs a derivative expansion in the target space ge ′ α √ ], and emerge from the four-loop In either approach, for a generic target space it is prohibit The two ways of obtaining target space field equations are of c 8 4.1 Target space4.2 diffeomorphisms Worldsheet diffeomorphisms string length the exact string equations of motion. Rather, one typically in [ equivalently a loop expansion parametercurvature in corrections the were first worldsheet seen n from the point of view o the corresponding tree-level scattering amplitude of grav are the Einstein equation together with equations of motion worldsheet conformal invariance requireswhich imply the the vanishing target of space fields obey certain equations of energy behaviour of strings. series of higher-order corrections play an important role i around the background. The linearizedvertex field operators equations have arise the correctnon-linear anomalous field conformal equations weig are the condition for vanishing w of the same thing. Perturbatively,deformations vertex operators of in the the worldsheet w background action, and (at correspond least toadmissible, in for the fluctuation massless it describes states). must obey In the targe orde also be captured atarbitrary the curved non-linear background, level composed by of considering a the metric w 1 Introduction It is a highlyenergy non-trivial limit of (if closed well-known) string fact theory.matrices This of that equivalence the General was two fi theories: the 5 Supergravity equations of motion6 as an anomaly Conclusions 3 Curved target space: classical4 aspects Quantum corrections Contents 1 Introduction 2 The flat space model JHEP02(2015)116 s ]. In 15 currents , chirality. 14 corrections. ′ ] whose spec- -systems have corrections — same α ′ we examine the 13 such amplitudes βγ α – 4 ]. Strikingly, they we present, at the 11 ]. 19 3 why – 20 by briefly reviewing the 17 sed [ or as a complexification 2 1 e fields vanish so that the arget space and worldsheet. o massive modes in the the- in the flat space model was ; there are no upersymmetry in supersym- t, the action we find is a type e is a formulation describing acquire quantum corrections. supergravity amplitudes when free, opening the possibility of nlinear supergravity equations azo, He and Yuan [ 3 nation of his theory is chiral, and may be rings, ns — with no ns’. These equations were known maps into a curved target space. The appropriate generalization is exactly ng theory. In type II string theory, the two ], it is not known how the fermionic worldsheet he bosonic portion of this theory can be obtained as in this model they are of the 11 ], and indeed the chiral worldsheet theory 16 – 2 – -point correlation functions of vertex operators ] is that the supergravity amplitudes appear di- n energy, fixed angle regime [ 11 ]). These currents are gauged and, as in flat space, 22 ], and are rather subtle. In section high , 32 21 – 23 -system. The quantum properties of curved βγ ] describes maps into flat space-time and computes amplitude ], pointing out its key features. In section 13 11 – conditions for quantum consistency of such a model. 11 we show that the algebra generated by the quantum-corrected 5 exact -point tree level amplitudes in supergravity ] is closely related to the twistor string constructions of [ n 12 , This paper provides such a description. We begin in section The theory in [ Recently, a new first-order worldsheet theory has been propo A salient feature of the model of [ This interpretation is, at present, only heuristic: while t 11 1 been extensively investigated [ should be the fields can be foundsets from of an worldsheet infinite tension have limit opposite of chirality, type where II stri Finally, in section making exact statements about itsof quantum behaviour. supersymmetric In curved fac behaviour of the currents underWe learn diffeomorphisms that of the both classical the curved t space currents of section responsible for localization onclosely the related scattering equations. to themetric quantum Hamiltonian mechanics framework (cf., of [ at worldline genus zero s itcurrents is disappear possible from to the choose action. a The gauge remaining action in is which the gaug worldsheet theory of [ classical level, a generalizationThe of key this is model to describing generalize the worldsheet current algebra that also govern string scattering in the perturbatively around flat space.maps into It curved space-time. is Since natural thelinearized theory to around produces ask pure flat if space, ther the supergravity field equatio rectly in the remarkableparticular, representation the discovered by worldsheet Cach theory provides a natural expla of [ trum consists only of theinterpreted states either of (type as II) the supergravity.of infinite T worldline tension supersymmetric limit quantum of mechanics.ory type There and II are correspondingly, st n at genus zero, compute via a chiral infinite tension limit of the [ are supported on the solution setto of be the ‘scattering closely equatio associated with twistor strings [ is anomaly free if andof only motion, if the with target no space higher satisfies curvature the corrections. no JHEP02(2015)116 γ χ = , ¯ µ 2 c (2.1) (2.6) (2.4) (2.7) (2.2) (2.5) (2.3) ψ and . Thus ν χ P , ¯ ) and µ ) as usual. e ν ¯ ). All other ψ ¯ ψ , 2 , taking values 2.4 / . µν 0 ν µν 1 0 Σ 2 η η ¯ P G )–( ∼ µ ,T 0) SUSY algebra. + ) , ldsheet, while ˜ P := µ 2.2 (Σ w ¯ γ , ] for the pure spinor 0 ψ µν and ( 0 ¯ ( ¯ ∂ G 0 η 1 2 ¯ ¯ 12 0 ∂ǫ , β = (1 ¯ G , e 2 ¯ ǫ − H uge in which ) = 0 + ¯ ∂ N z and + ( µ 1 = s H − 0 d gh ν ¯ = 0. Likewise, the fermionic ψ µ ∂γ by ˜ c ¯ 2 ined from a first-order action G P g these transformations obey β = P tion, which becomes free ν g. See [ ], so it is tempting to interpret µ as µ P + ¯ ¯ χ + ψ ¯ 2 ∂α ψ 11 µ ction for a massless particle with 0 , ˜ c µ 1 ¯ P µν − G ¯ , ∂ ψ are conjugate to := , δχ b is a complex ˜ γ µν 0 are valued in Ω = ν χ χ η 0 η µ + ǫ + , δ P ∼ G ), respectively. Note that all the ghost ψ + 0 µ ) 0 limit [ := ¯ µν ∂c µ G and 2.4 b w 0 P and ǫ P ( γ → ǫ η 0 µ χ H + ′ ǫ )–( ] that describes gravity perturbatively around , δe = ¯ G = α µ : +¯ ν ) µ χ ψ 11 µ – 3 – 2.2 z P ¯ ) behaves like a Beltrami differential and acts as ψ ¯ ( ∂ψ for later convenience. + ¯ 0 Σ µ µν ¯ G ψ µ bc ∂c ψ ,T α η + ¯ ∂ψ (Σ + : into , δ µ µ = 1 , δψ , ¯ is conjugate to the gauge transformation 2 m , µ ψ 0 , ν µ 1 e ¯ ¯ Ω ψ ∂X w + ψ ǫ ψ c T 0 µ δX µ ∈ µν − P H = ¯ I e z Σ ǫ η µ ¯ ∂X Z = ) enforce the vanishing of µ ∼ = 2 π P / 1 δX ) Q µ 2 1 Σ ) is a (1,0)-form on Σ, while Σ w ( Σ Z ), while those imposed by ¯ = ,T 0 δX Σ ¯ π G 1 S ,K 2 (Σ ) 1 ,T , ). The field z (Σ 0 ( 2 = 0 / (Σ 0 was written in terms of two Majorana fermions Ω 1 0 Σ ). We have combined S Ω G ν µ Ω ∈ ¯ ψ ∈ ψ ,K ∈ is the usual ghost for holomorphic diffeomorphisms of the wor ] µν µ η (Σ c P α 11 are ghosts for the transformations ( 0 χ, χ − γ The constraint imposed by We can use the BRST procedure to fix the gauge redundancies ( µ In [ 2 ψ ( 1 2i spin. The terms involving onlyfor bosonic standard fields string can theory also by taking be a obta chiral the action is just a chiral generalization of the worldline a fields ¯ a Lagrange multiplier imposing the constraint fields remain invariant inthe each OPEs case. The currents generatin where and all vanish. In this gauge, the currents disappear from the ac and matter fields are purely left-moving. The BRST operator i In the absence of vertex operators, at genus zero we can fix a ga the theory as a chiral,version infinite of tension this limit model. of the RNS strin in ΠΩ flat space. In conformal gauge the worldsheet action is given where We begin by briefly reviewing the model of [ 2 The flat space model respectively, where the fermionic parameters ¯ where Note that the target space metric enters the model only throu which may be viewed as an infinite tension limit of the standar and JHEP02(2015)116 e X to ¯ · ∂c k the b i µ (2.8) P ) = 0 vertex z ( fields at n 2 . Closely µ µ e P P k = and just suffice 0 } H ] the NS-R, R-NS i k 13 { tems other than . In summary, replacing description allows one to g that it provides the origin l graph topologies for the path integral forces } menta orting the interpretation of ive excitations, which have formal weight (as would be 3 traint that rge from double contractions e further provided a general- in the BRST operator, rather X mensions. An important point − e zero mode of the -form fields that complete the 0 , localization onto solutions of sheet p external momentum -particle scattering of NS states H n OPE is trivial, the plane wave e ] for tree-level scattering amplitudes vanish automatically at genus zero. and , . . . , n 15 2 1 0 , ¯ P G XX 14 , ∈ { 0 G and ] for details) and leads directly to the formulæ µ -field and dilaton, while in [ 13 vertex operators in this theory correspond to – 4 – P , , i ) becomes meromorphic with simple poles at the B 3 z ( 11 ) valid at arbitrary genus µ = 0 P 2.8 j j z k · − i i k z i 6= likewise becoming a meromorphic quadratic differential. Th j X is nilpotent provided the space-time dimension is ten, as in 2 ], NS-NS sector P Q 11 . 2 , ). again vanishes everywhere. The worldsheet correlator of th 1 ψ 2 2.6 P description of supergravity by a chiral world 3)-dimensional system of equations is the holomorphic stress tensor for all matter and ghost sys ] we went on to provide a similar formula for line − m 13 n T ]. As explained in [ In [ The main claim to fame of this description of supergravity is With respect to 14 3 finite ( insertion points, with When vertex operators are inserted, be holomorphic over Σ, and hence both identically over Σ. In the absence of vertex operators, the than from requiring thatthe they case have in the usual correct string anomalous theory). con Indeed, since the trade the problem ofproblem computing of amplitudes finding by solutions summing of over the al scattering equations. usual superstring. always has vanishing conformal weight, irrespective of the the world of [ involving arbitrarily many NSthe sector scattering states. equations arises In as particular a consequence of the cons perturbations of the target spaceand metric, R-R sectors were shown to provide the gravitini and at genus one. The loop integral arises as the integral over th of the striking formulæ of Cachazo, He and Yuan [ spectrum to linearized Type II (Ais or B) that supergravity the in ten linearized di between field the equations vertex on operators and the the target currents space eme where genus one, and is expectedthis to expression diverge. as We gave the argumentsization one supp of loop the integrand scattering of equations supergravity. ( W following from ( related to this iseffectively the decoupled fact in that the infinite the tension spectrum limit. contains no mass fix the moduli of this pointed curve in terms of the external mo operators can be simply computed (see [ to ensure that JHEP02(2015)116 and (3.2) (3.6) (3.4) (3.5) (3.3) (3.1) M T ∗ X neralization of the ) remains invariant. s µσ 3.3 σ νρ ψ rved target space. In this R ) its complexification with λ λ σ is the holomorphic tangent ¯ es ψ ψ 1)-part of the pullback to Σ µ ψ al and will be investigated in , κ rial transformation ations are generated by the currents at the classical level. ρ ν σ Π ¯ M , , ψ ¯ M, g ), the natural generalization of µ ˜ ψ T X µ µ ν ψ µ νλ ∂ ∂X ¯ ψ  ¯ Γ χ, e , ¯ ν . The presence of the Levi-Civita ¯ ǫ ∂ψ λ = 2 Dψ ν g ˜ µ κν λ ψ µ χ, X  ¯ is the (0 ψ − ¯ ψ κ ψ ∂ λ where is the Riemann curvature of the Levi- X ǫ g ¯ λ , since all the fermions are left-moving. ρ  ψ µ 2 ψ ¯ + σ cl + ψ ˜ C κ ∂ X κ µλ µ S ¯ µ Π ) is ¯ ψ ) + νκλ ∂X ∂ Γ λ µν λ ν µ λ ν , so that classically ( → ¯ ¯ κ µλ ψ ψ ¯ R ∂X ψ − ¯ ψ + ∂X κ Γ M µ κ M, g µ + Γ ¯ µ M ν + µ νρ ψ ¯ – 5 – ν ψ Π P T − µ Π ν Π ¯ ) is invariant under the transfor- ψ λ µν 2 Σ P κ νλ Σ  µ Π ν µ ) of Γ + Γ Z Z Γ µν ν σ Π ˜ 3.3 ¯ ¯ X X µ ψ := ψ κν π  π ( 1 − 1 ∂X ǫ g ∂ ), where 2 2 . Notice that, unlike the standard string, here it is not µ µ µ µν 2 ν ¯ : Sym ǫ g ∂ψ / ˜ − ψ g Π ν ρµ = X = 1 = M Σ g − µ (Π Γ = µ cl cl := := µ ,T 7→ ˜ µ µν ρσ Π S S ¯ ǫ ψ Π cl cl µ are now understood to take values in the pullbacks (Σ ¯ ǫ ǫ g ǫ g − G G ¯ 0 Dψ X ¯ 7→ ψ = = ¯ = = µ . The action ( ΠΩ µ µ µ µ 0 ¯ Π ψ and . That is, Π ∈ δ δψ H δ g δX ¯ ǫ ψ ǫ, and ) be a pseudo-Riemannian space-time and ( R 0 ¯ G , g . Temporarily ignoring the gauge fields ( , R 0 M M G , respectively, while ∗ M We may further simplify the action by introducing the field Π a Let ( The target space metric does play a role in the curved space ge T ∗ whereupon the matter portion of the worldsheet action becom Civita connection.Noether At currents the classical level, these transform of the Levi-Civita connection on possible to include a four-fermion interaction in under the diffeomorphism with parameters connection in the definition of Π is reflected by its non-tenso and does not depend on the choice of target metric X currents mations where the fermions We now seek to generalize thesection flat we space confine model ourselves to toThe a the quantum discussion case mechanical of of behaviour the a of cu actionthe this and following model section. is non-trivi 3 Curved target space: classical aspects the matter action for the case of a curved ( holomorphic metric bundle of JHEP02(2015)116 µ cl ut X H (3.9) (3.8) (3.10) ). Note . They 0 , 3.7 H  σ . νκλ ntization ¯ ψ ν ρ H and . As in the flat ¯ ψ µ ψ 0 µ ¯ ∂α ¯ ∇ G ψ νρσ λ − λ , = 0 (3.7) ¯ 0 ψ cl H ψ = o κ G κ 1 2 , then the currents are H ¯ cl ψ ψ e ¯ B 2 ν δe G + , µν the flat space model to a ψ σ and Hamiltonian in super- ese currents, with local pa- + the Levi-Civita connection. cl µ um theory. same algebra sappear from the action. s it is possible to choose the κλ = d ¯ ψ ¯ urrents still obey ( G netic terms in the action. In ψ cl and ρ mensional space of maps from R ¯ n ¯ G 1 ψ 3! H 2 1 is canonically conjugate to χ ¯ ∂ǫ ρ νσ − µ − − + Γ ) , cl = σ . As in the classical string, including − νκλ G 2 ψ ν χ ρ Z H δχ = 0 Π ¯ µνκ µ ψ , + ¯ ¯ ǫ o H ∇ ρ νσ µ ¯ ∂ κ cl   λ Γ ¯ ¯ ψ G λ − ψ ¯ ∂ψ ν , κ − ψ ¯ µ – 6 – ψ ¯ = κ cl ψ ¯ ν ψ µ ν G ψ ¯ χ ¯ ¯ ψ ψ δ n + )(Π µ 1 3! µκλ µ λ ψ ψ H + 1 3! κ ¯ 2 1 ∂X , ¯  ψ µ λ µνκ − cl + appear simply as the torsion of the connection, but rather Π κ µλ ψ ν , with 3-form field strength H λ H κ Γ Σ ψ κ ¯ ψ Z ψ M µ = vanish the worldsheet action is free and the theory knows abo κ ψ − not ¯ ) thus generalize the flat space currents ψ ν π κ µλ ψ 1 µ χ o λ 2 ψ Γ ¯ cl κ µλ ψ µ 3.8 ¯ (Π G κ = Γ − ψ , ¯ and ¯ ψ µν µ S 1 cl − 3! g χ generates target space Lorentz transformations, after qua -field on µν Π G µ respectively, are gauge symmetries of the action )&( ,  + ν B n e κλ Π := } ν ψ µ 3.6 ¯  R cl µ ψ -symmetry of the fermion system to Π ¯ 2 1 ∗ ψ µ H µν µν C ψ g g -field here does − ¯ ǫ, ǫ, α = { B = = = ν . µ cl cl cl J ¯ M G G H If there is a At the classical level, the transformations generated by th without changing the action. The Poisson brackets of these c further modified to that the is a Lichnerowicz Laplacian actingΣ on forms to on the infinite di provided the gauge fields transform as The currents ( The Poisson brackets of these curved space currents obey the where the equality in the first line follows by the symmetry of a target space dilaton is best done in therameters context of the quant breaks the as in flat space, where now take a similar formsymmetric to quantum the mechanics. worldline supersymmetry In currents particular, since Π 4 Quantum corrections In the previous sectioncurved target we involved showed changing that thethe gauge currents, the where but generalization not of the ki space model, at genusparameters zero, so in that the the absence gauge of fields vertex vanish operator and the currents di while JHEP02(2015)116 ] ) ) or 27 w 4.2 – ( (4.4) (4.1) (4.2) (4.5) (4.3) ν along ¯ w 23 ψ V ν − L V z µ , ∂ will see that ) as they take is that it does try −  ) M -systems [ z ∼ ( ) µ naive V vel. The quantum βγ ]. The first piece of w . V O ( properties of the cur- µ 32 µ w – or ∂ ) transformations of all ¯ ν by the Courant bracket ψ µ ǫ 1 − ) δ 29 M urved ) does not agree with ( M z z − ( T w e [ ) ( V  to have the OPE gent bundles. gher curvature corrections — ∼ : , ], though the supersymmetric z ( O lly under Diff( ) ν ) iant at the quantum level. , W µ r: the curved space version of a 27 ions may be computed using the ψ to represent the diffeomorphism w t on w d by the Lie derivative naive W ) – ( ( O µ , ν ¯ w O V ψ  23 ) ( ¯ )) Π : : ) ψ w ] ǫ O λ w z ) and ( ( µ − ψ z + ν w ( κ V V,W z V -systems in general. It also sits harmo- [ z ψ µ ¯ ν ψ ( − O naive : µ V ∂ O βγ z X O V κ ( ∼ ν : + µ V ∂ ) µ ν V – 7 – , ψ w Π ∂ ∼  ( µ µ w 0 ) ∂ ν W V µ → w : − lim ǫ δ O we require ( : + ) z µ ν z − ψ . ≡ ( W ∼ Π ) : ) and generates the correct Diff( V z ) ν := µ ( O w V 4.2 V V and ( Φ) only through the BRST operator. The resulting action is )Π µ ν O system O ∂ - V X : (  )Π µ , βγ g, B, z -systems is known to be subtle [ w V ) ( : µ w 1 w − βγ . To realize this in the quantum theory, we must seek an operat ( X − − µ z V z V curved ) := ∼ − ∼ z ) ) ( ] is the Lie bracket of the two vector fields. One might naively . In our supersymmetric context a further problem with w w ∗ M ( ( ) under diffeomorphisms of both the target and worldsheet. We µ µ T naive V V,W X ⊕ 3.9 O )Π ) z z M Remarkably, these two problems cure one another. The operat In this section we use this OPE to examine the transformation We now examine the properties of this theory at the quantum le that generates this diffeomorphism. In order for ( ( T V V V O O values in the pullbacks of the target space tangent and cotan both obeys the desired OPE ( rents ( not act on the fermions, whereas these transform non-trivia should be solvable. niously with the motivation forworldsheet this theory paper describing pure explained supergravity earlie — with no hi This is one of the main advantages of curved on where [ because of double contractions.whose resolution This usually is requires replacing a the common Lie feature bracke of c free OPEs case is much more straightforwardgood than news the is that purely since bosonic the on action is free, correlation funct but this fails for two reasons. Firstly, the OPE fields. That is, we have the OPEs the target space fields ( behaviour of curved an example of a Infinitesimally, target space diffeomorphisms are generate these currents must receive corrections in order4.1 to be covar Target space diffeomorphisms some vector field O algebra, given two vectors JHEP02(2015)116 , cl cl G G . As (4.6) (4.8) (4.9) (4.7) V -systems . ¯ βγ G transform as , } ¯ ψ ¯ w and G V G − ), this is not true L , ψ, z M Π xpected non-tensorial involve (holomorphic) + , derivative along , we simply choose Ω to is the Levi-Civita covariant X, , contains a non-vanishing { νρ µ r target space diffeomor- e currents as M e contraction between e quantum level. We em- sical level the currents ∇ V ··· ∂g ∼ · · · ibutions to the higher-order pears to be due to the new, O m parts of ) uarantee that the same is true + µ νρ mentation, one finds that the rsymmetric curved w or higher) contractions between ) Γ ) must be modified in the quan- . ( ν µ ¯ G ¯  ψ ψ , where 3.9 with ancellations between terms arising ) µ µ κ µν z + w V ( cl  Γ V mplex space, but may be expected to lead to µ  G V κ ν − κ µκ , ∇ ¯ ∂ ψ O z Γ µ  µ ∂ ∂ ( log Ω ψ Ω = ) than their bosonic counterparts has been ν ∂ µν V ∂ ∂ g log Ω M µ L instead. + – 8 – ¯ 1 µ ψ ∂ − d : + : + µν µ x , g ψ cl cl d ¯ L L G G w ∼ · · · G  ) V − ∂ ∂ ) is the pullback to Σ of a top holomorphic form on the = : = : w L z , and also modify the coefficients of the simple poles by ( d log Ω = Ω ¯ G G cl x V ) transform geometrically under Diff( : + : + ∧ · · · ∧ + V d G O 1 cl cl L ) To incorporate a dilaton field Φ on ¯ ) ensures that the fundamental fields 3.9 x ]. G G , z d ( 4 V . 32 4.4 g V ∼ · · · – = : = : ) √ M O ∧ · · · ∧ ¯ G G 29 w 1 2Φ ( x − G d ) g z ( √ V ( ∗ introduced in ( O X cl H . V O In order to include a dilaton it is convenient to rewrite thes Although the choice ( To correct this anomalous behaviour, the currents ( and the composite operator. In particular, while at the clas and Thus, for any vector field 4 V cl ¯ noted before, see e.g. [ first-order pole behave more straightforwardly under Diff( G (complex) target space where Ω = and so are covariantphasize under that target while space thederivative diffeomorphisms net terms, at transformation this th of calculationfrom the involves double currents non-trivial contractions ap c from both the classical and quantu tum theory. Theworldsheet required derivatives. modification Such ispole terms to terms generate add in both new the new terms OPE contr that with be the pullback of e in the quantum theory. For example, the OPE of of composite operators because ofO the potential for double ( expected under target space diffeomorphisms, this does not g behaviour of the Levi-Civita connection. The fact that supe where again the second term in the transformation of Π is the e terms involving worldsheet derivatives.modifications should After be some experi and These quantum currents dophisms, having indeed the behave OPEs appropriately unde derivative. The existence of Ωinteresting is constraints not on restrictive possible on compactifications. an affine co with all quantum anomalies, the origin of this term is a doubl which does not combine with other terms to form any sort of Lie JHEP02(2015)116 : 2 ) are ], and (4.11) (4.12) (4.10) 4.7 35 ], the tar- diffeomor- ensor by a likewise be of ( 33 ¯ , G 7 and ) is well-known in G to vanish locally in heory [ worldsheet 4.12 Σ ] reviewed in section  affect the condition for : R ir potentially anomalous rldsheet stress tensor µ 13 ized field equations came ¯ ). Thus, despite the pres- e short distance OPE, and ) OPE is trivial, so there ψ , , w ∂ , w rrents. ( 11 µ ( n T-duality, see e.g. [  . T ··· ψ ) transform covariantly under f the quantum theory to be g X )  ) + z √ g , the currents z 4.7 ( ( √ on coupling ( : + : 2Φ T 3 X µ − ) under worldsheet diffeomorphisms. 2Φ e ), and so are not related to worldsheet log Ω w − 2 ∂ψ e / w µ 3 ( Σ µ − ∂ ], in particular the fact that the dilaton ¯ µ log ψ T z K -corrections to string theory using doubled : ψ ( ′ 34 ) log Σ z L α , ( 2 1 2 R 1 2 ∂ – 9 – T 26 Σ − 2 1 Z : compared to the usual dilaton coupling in string µ ∼ − π − g 1 ) OPE (including ghosts) is the critical dimension 8 ) cl ). For example, there is now a triple pole in the OPE w √ ( ∂X w T + ( µ T 3.3 G log S ) := ) 2 1 z :Π -field and dilaton on the target, the only restriction on the z ( G → ( T − T − B cl S T Φ := cl → T ) of stress tensor implies that the worldsheet action should ]). 4.11 is no longer primary. The resolution is to modify the stress t 36 G is the worldsheet curvature. We can always choose Σ ) = 10, as in flat space. In particular, unlike in usual string t ) transformations, they also affect their behaviour under R M ( M The choice ( Note that unlike in string theory, this modification does not C -functions. This is as expected from the flat space theory [ dim It is straightforward to check that using this stress tensor ence of a non-trivial metric, β model to emerges from the geometry (cf., [ phisms. This can be seen by considering their OPEs with the wo are no new contributions to the fourth order pole in first-order formulations of stringis effectively theory shifted [ Φ analogous shifts also appear when studying modified to the requirement thatnot the from vertex any operators anomalousbehaviour had conformal under to transformations weight, generated obey but by rather linear the from gauged cu the get space field equations do not appear in primary operators, transforming as sections of between the stress tensor and Diff( two dimensions, so theour addition calculations of are this self-consistent. term does Actually, not the affect dilat th that follows from the free action ( theory. This shifted coupling also plays an important role i where While the quantum corrections ensure the currents ( 4.2 Worldsheet diffeomorphisms worldsheet conformal invariance, because here the showing that total derivative term; that is, we choose the stress tensor o JHEP02(2015)116 ′ . ), α M as a 4.7 (5.3) (5.4) (5.2) ′ α . ) OPE has w ( λµν Φ) ¯ G ν H ) ∂ κ z µ w ∂ ( ∂ .) Hence the only ν G ν g − = 0 (5.1) ¯ ψ ψ at least locally on z und, treating √ . µ µ o w ¯  cl ψ ψ tisymmetry of fermions . B , and so obtain the exact ) ( ¯ λ − G log ρ consistency of the current tries of the Riemann and ¯ , ψ ∂ z ν ce fields. ble, interacting worldsheet κ cl ∂ = d the quantum currents ( = 0 ¯ νσµ e find ¯ ψ µ G evel. Contrary to usual string n the target. We have no evi-Civita. So neither of these variance places no restrictions ∂ ) R s + 2 n H 3 1 currents is non-anomalous, will ρσ exactly ) µν + m the usual case in string theory, ( = − ρ κ νκ R  Γ κ νκ , ) that behave correctly under both µ µν µσν Γ w ∂ ) is given by the first term. This is µ R R 4.7 w ν ( ∂ − = 0 ν ψ ¯ 5.2 z − ψ ρ κσ µ o µ z cl ψ ¯ ψ ( G + Γ = 0 and the Riemann and Ricci tensors obey ∂ , = 0 and is closed so that ∂ κ -field. ] cl – 10 – H − B G + µν H [ σρµ n R λµν κλµ H ν ρσ ν ) OPEs are non-singular, while the ,R ) OPE. Again performing all possible contractions and κ = 0. It is a remarkable fact that because the worldsheet Γ R w , ∂  ( w 2 : ) OPE. Performing all possible contractions and expanding ( cl ¯ ν κ G ν Q ¯ = 0 G w ) ψ H ψ ] ( ) z µ w µ w w ( ∂X z G = ¯ ψ ( ψ ¯ ) ν G κλµ − λ − ¯ [ − λ G ¯ z o ν ψ ¯ z ( ψ z ψ z ) obey cl µ R κ κ G ¯ ¯ G ψ ¯ ψ ψ , 2.7 : ) and 1 3 cl 2 1 w + 2 G ( n ∼ G ∼ − ) ) ) z w ( w ( G ( ¯ G G ) is indeed the field strength of a ) z z ( ( ¯ H G G Instead, the target space field equations arise from quantum We begin with the We now turn to the only a simple pole. Onlythe if this BRST is operator true, ( so that the algebra of the identities These anomalies vanish provided again d the coefficients of higher order poles around the mid-point, w algebra. At the quantum level, the Poisson bracket relation loop expansion on the worldsheet, or derivative expansion i between the classical currents should be replaced by OPEs of These are of courseRicci the tensors that first hold Bianchi providedtwo the identity OPEs connection and Γ impose is basic any indeed symme dynamical L restrictions on the target spa non-trivial anomaly cancellation condition in ( simply the requirement that the 3-form so that the target space and worldsheet diffeomorphisms attheory, the the quantum l requirement of quantumon worldsheet the conformal in target space fields. CFT and so must work perturbatively around some fixed backgro action is (locally) free, we canquantum compute consistency these conditions. current This OPEs iswhere quite for distinct fro generic backgrounds, one is faced with an intracta The second and thirdcontracted into terms partial in derivatives. this expression (Recall vanish that by the an parameter. In the previous section we constructed currents ( 5 Supergravity equations of motion as an anomaly Thus expanding around the mid-point we find JHEP02(2015)116 ρ ψ (5.5) (5.7) (5.6) as its anish. λ σρ uantum Γ H  λ κσ ψ g κ µν σ  ) Γ ψ = 0, which in ν λ ρσ ψ ∂ λ H σλρ H µν y if the space-time -field and dilaton. κλν H g . ) OPE has B λ H µρσ + ( w w ). This OPE has first,  ( ρ H ) ∂ ∂ψ = 10 supergravity field  ¯ w H o obey 4 1 G − κ 2 ( , . , σ νρ ) ¯ + d 2Π pole defines the quantum ¯ z ψ G z − H   ( (Γ − + λ 1 Φ ∂ = 0 G 12 um corrections to the curved  ρ µν λ  ψ y with a 2 y, calculation yields ρ µσ Φ = 0 Φ = 0 Φ − ∇ 2Φ )Γ ν ) and κ Γ H κ µ µνλ z − Φ ∇ ∇ µν ( e 1 ∇ µν ∇ 12 µ H µ g g G g κ ( µν  µν ∇ ∇ κ ) appear as the coefficients of higher √ − ∂ + H ∂ + 2 Φ 1 2 2 H κ Φ µ  5.7 2 + 2 ¯ µ µλ ψ + − log ∇ σ µν R − ∇ 4 κλ ν ν Γ κ µν µνκ  ∂ Φ σ µνρ H µν − µ µ κ H ∂ . H νλ ∂ H κ 1 2 Φ 1 2 ∇ 2 – 11 – i g H ) µ µκλ 4  ∇ κ ∂ ) − µν + Φ) obeys the non-linear ν ∇ Φ H ∇ − g ¯ ν µ ψ 1 4 µνρ ( ρ σρ  ∂ µ = 0 at every point of the worldsheet. As reviewed in 2 Φ ) ∇ Γ 2 H µ ∂ − ν ψ takes the somewhat unenlightening form 2Φ ν κ µλ ) g, B, ¯ 1 2 σ µν ψ 1 ; − ∇ Γ Π 12 w + + 4 2 µν µ e H Γ µ µ ) ¯ − g µν κ M 2 1 ψ R − R − ∇ Π w g  √ ) z 2  − + ( µν − ∂X 3 ν + 4 η σµ − ) z σ µν µ κν g ψ ( log log Ω ( R w Φ µ ν 2 ∂ ν (Γ λ ψ ΦΓ ∂ − ∂ ρ ( σ µ ∇ µ z ψ κ ∂ Π ∂ , but the coefficients of the higher order poles must be made to v ( ν + + 2  condition for the worldsheet theory to be consistent at the q  ∇ µν H ψ ) ∂ g ∼ ) λ L ) µν µνρ ψ g w µν κ ( ( ∂ H g exact ¯ ¯ Φ) obey the equations ( ∂ σ ψ G +  h ∂ ∂ ) ) we see that the algebra of currents is anomaly free if and onl z 2 4 1 1 , this the content of the scattering equations. The ∂ ∂ cl ( 2 5.5 g, B, H G − − − − The BRST operator constrains physical field configurations t The only remaining OPE to be checked is that of = H equations, in the Neveu-Schwarz sector. flat space is the condition section level is that the target space ( classical contribution, while the field equations ( Hence, the The quantum corrected current Proceeding as above, a straightforward, if somewhat length These are precisely the field equations of general relativit poles. In this sense,space the generalization Einstein of the equations scattering emerge equations. as quant fields ( second and third ordercorrected poles. current The coefficient of the first order From ( JHEP02(2015)116 . ) ) ∈ 2 = w − ( 5.5 ) 0 ¯ (6.1) Π) G arget w H · ) z k − ( ( z G ¯ δ equation of ) OPE ( , Φ around flat w ν ( δ on the marked ψ ¯ G µ e erator describing ) or ψ = 10 supergravity. z λ ( d ¯ G ψ δB ring equations κ to linear order around ¯ ψ heet theory encodes the ) the be worldsheet instantons H ra in string theory. The pears at higher loop order around flat space (and re- ented here. Perturbatively, vertex operators are found κ νλ the exact target space field scribes maps to a (complex- eyond what can be seen in a ¯ Γ tions of G . supersymmetry-like transfor- δ e vertex operators in different e the BRST operator is nilpo- ( -system, the path integral has on limit, despite the worldsheet ctuations e the vertex operators arise by µ ∂ lly, it would be fascinating to un- and ons provide a natural example of βγ − G , and so is free. The theory involves λ -field, and dilaton around flat space. ψ M B κ = 10 supergravity equations of motion, ¯ ψ ]. Non-perturbatively, we must remember -field equations enter at order ( d B κ νλ 37 Γ δ µ – 12 – Π -field equations, so that the triple pole in µν B η 2 ). The worldsheet action is a type of supersymmetric − ν Π M, g µ Π µν . For example, expanding the metric in δg , the remaining factor of the integrated vertex operator is = -functionals in usual string theory. Of course, the dilaton µ 0 , whereas the Einstein and k β 3 ] for details. − around flat space. When the fluctuations are plane waves with t ) 11 currents w δg scattering equations presented in [ H − H − ), which is best interpreted as a modulus of the gauge field -system. Unlike for a purely bosonic curved = 0. This quadratic differential is essentially the vertex op z Σ ν βγ ,T Π µ (Σ Secondly, note that the dilaton equation of motion enters in We close with a few remarks. Firstly, the curved space worlds We briefly consider dimensional reduction of the theory pres massive 1 Π , 0 µν expressing them in terms‘pictures’. of See real [ fermions) likewise gives th curved space are obtained similarly. Expanding the currents that the theory herewrapping lives holomorphic on curves a in the Riemannpurely target. surface, worldline description These and of effects there supergravity. goderstand can More b whether genera D- can survivetheory in this being infinite chiral. tensi at order ( worldsheet. The integrated vertex operators describing flu This is analogous toin the the way the worldsheet dilaton equation of motion ap vertex operators for perturbations of the metric, In this paper we haveified) constructed Riemannian a worldsheet manifold theory ( that de 6 Conclusions H In the non-linear sigmaby model considering of linearized string perturbations theory,perturbing the of the flat the space action; her no anomalous behaviour under diffeomorphisms of η fluctuations mations, and iscorresponding a algebra chiral of analogue currents istent, of anomaly if the free, and supersymmetry and only henc algeb if the target space obeys the gauging a certain worldsheet superalgebra which generates space momentum with no higher curvature corrections. the Minkowski metric one finds up to terms which vanish on the support of the flat space scatte motion is implied by the Einstein and is guaranteed to vanishequations if indeed the arise double from poles a do. 1-loop anomaly In of this the sense, currents we expect that this should correspond to Kaluza-Klein reduc the Amplitudes involving scattering of Kaluza-Klein excitati JHEP02(2015)116 ) c M ,X ]. It 41 = 0 and , H 40 = 0 exactly. equations as . It would be 3 H CP × , often known as the 3 he worldsheet action. ], Berkovits presented M ]. Indeed, this was one = 0 is solved by writing CP 12 2 43 ibe an ambitwistor string , viewing the theory as a how the considerations of derstanding of the twistor s. This seems particularly the vertex operators which s provide a natural origin P cult to understand how to directions over any Cauchy hic) cotangent bundle of are naturally interpreted in een explored in [ anics, or as a chiral infinite M tors corresponding to target e in ng the constraint so useful. The fields (Π ]. However, it has the usual lobally hyperbolic space-time t give a simple example of the the constraint space-times where the Cauchy int H string for pure SYM is a long ed perspective. . The relation of the flat space 14 ] for scattering of gluons to the . It is important to understand l rays [ aces. amounts to taking the symplectic 2 14 and H solve ¯ G ] between twistors and the pure spinor , ]. G 42 11 and (once compactified and projectivized) ˜ λ λ – 13 – = P ]. 19 , ]). Secondly, and perhaps more importantly, the ambitwistor 18 39 ]. Adding the worldsheet fermions provides a supersymmetri 38 -point CHY amplitude formula [ [ n M . The resulting space is the space of null rays in H , and in terms of the interpretation of the supergravity field ′ by α ∗ M T Finally, it would clearly be very interesting to understand Throughout this paper we have used the ‘RNS’ formulation of t For some space-times it may even be possible to There are two main advantages to this perspective. Firstly, In this paper we have concentrated on the geometry of terms of cohomology classes onPenrose ambitwistor transform space, (see and e.g. in [ fac integrability of a certain superconnection along super nul fascinating if the curvedand model ambitwistor here models could of [ lead to a deeper un together describe a map from(with the Π worldsheet twisted to by the the (holomorp worldsheet canonical bundle). Imposi the ambitwistor space can be viewed as a quadric hypersurfac this paper can be(super-)Yang-Mills adapted equations. to relate Finding thestanding an CHY problem. (ambi-)twistor formula [ a pure spinor versionhow of to the generalize flat the spaceimportant, pure model both spinor in of model the section to lightstring of curved at the background finite close relation [ of the original motivations for considering ambitwistor sp This has theof advantage the that Pfaffians the indisadvantage worldsheet of the fermionic the RNS spinor formulationspace that gravitinos while and vertex form operaturn fields on can nonlinear be background constructed, fields it beyond is the diffi NS sector. In [ surface is chosen tomodel limit onto to (past descriptions or ofwould future) be null gravity interesting infinity living to at revisit null these from infinity the has present, b curv space has several otherthe representations. space For of example, null rays insurface. may a be An g identified important with special the case bundle is of in null asymptotically flat version of this space.theory, which Thus was the the model point can of also view adopted be in saidhere [ to are descr treated as deformations of the currents quotienting by the gauge transformations generated by complexification of worldline supersymmetrictension quantum limit mech of the superstring. Another perspective is al the momentum as a simple bispinor For example, in four dimensional flat space-time the constra ambitwistor space of quotient of JHEP02(2015)116 , and , ommons = 1 (1981) 85 = 1 N N (2014) 017 134 (1974) 118 (1986) 409 , 03 , (1971) 222 irasoro model B 81 B 277 JHEP B 31 , amme (FP7/2007-2013) / s. The work of TA is sup- ]. n and International Trust. Annals Phys. , , redited. -function for the ant (FP/2007-2013/631289). , Cambridge. The work of EC β the European Research Coun- Strings in Background Fields Nucl. Phys. , SPIRE , Nucl. Phys. IN -model Nucl. Phys. [ , σ , ]. The Background Field Method and the ]. Four Loop Four Loop Divergences for the Torsion and Geometrostasis in Nonlinear (1980) 846 SPIRE SPIRE IN IN [ – 14 – ]. ][ ]. ¨he GeometryK¨ahler and the Renormalization of ]. D 22 ]. -model in Two-Dimensions σ SPIRE SPIRE SPIRE IN IN [ SPIRE [ IN [ (1985) 630 ), which permits any use, distribution and reproduction in IN ]. 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