On the Null origin of the Ambitwistor String Piotr Tourkine, University of Cambridge, DAMTP Dec. 6th, 2016, QCD meets Gravity, Bhaumik institute Based on: Eduardo Casali and Piotr Tourkine. \On the Null Origin of the Ambitwistor String". In: JHEP (2016). arXiv: 1606.05636 and work in progress with Eduardo Casali and Yannick Herfray. 2 / 30 Outline Motivations CHY, scattering equations Ambitwistor string Loops on sphere Tensionless (=Null) strings Classical analysis Quantization Relation to tensionful strings (Connection with Gross & Mende) 3 / 30 Motivations 3 / 30 Magic: X F = n-point field theory amplitude Jacobian solutions (gravity, Yang-Mills, scalar, etc) Cachazo, He, and Yuan 2014: breakthrough ! Z n Y dzi δ¯ sc. eqns. F (ki ; j ; zi ) i=4 • Scattering Equations: degree (n − 3)! 2 polynomial eqs. zi ki (ki = 0) • Theory-dependent part. i ; kj = polarisations, momenta 4 / 30 Cachazo, He, and Yuan 2014: breakthrough ! Z n Y dzi δ¯ sc. eqns. F (ki ; j ; zi ) i=4 • Scattering Equations: degree (n − 3)! 2 polynomial eqs. zi ki (ki = 0) • Theory-dependent part. i ; kj = polarisations, momenta Magic: X F = n-point field theory amplitude Jacobian solutions (gravity, Yang-Mills, scalar, etc) 4 / 30 Ambitwistor string Mason and Skinner 2014 Chiral (holomorphic) string living in Ambitwistor space. X 0 Z ¯ P S = P · @X 2 X P = 0 • Spectrum: type II SUGRA in d = 10 (there is no α0 at all). Same as low energy spectrum of string theory. • Scattering equations from P2 = 0 • Extend to higher genus amplitudes,1 + + ... 1Adamo, Casali, and Skinner 2014; Adamo and Casali 2015. 5 / 30 Scattering equations & Gross-Mende 2 X ki · kj P = 0 = 0 Fairlie and Roberts 1972 zi − zj j Govern the saddle point of the the Gross-Mende α0 ! 1 of string theory. Why do the scattering equations have to do with both α0 ! 0 and α0 ! 1 ? 6 / 30 Loop scattering equations on the sphere Our sphere prescription succeeded at one and two loops2. field theory propagators −! Where do these formulae come from really ? How to generalize them ? 2Geyer, Mason, Monteiro, and Tourkine 2015; Geyer, Mason, Monteiro, and Tourkine 2016a; Geyer, Mason, Monteiro, and Tourkine 2016b. 7 / 30 But QFTs are UV divergent in general, so how can this be ? Higher genus; modular invariance etc Z X 2 D −Im(τ)(`2+m2) 1-loop partition function ∼ d τd ` e i i τ Field theory integration domain i UV -1/2 0 1/2 8 / 30 But QFTs are UV divergent in general, so how can this be ? Higher genus; modular invariance etc Z X 2 D −Im(τ)(`2+m2) 1-loop partition function ∼ d τd ` e i i τ In string theory, modular invariance removes the UV region. i τ ! τ + 1; τ ! −1/τ UV This property has been assumed to hold also in the Ambitwistor string, which describes only -1/2 0 1/2 massless modes. 8 / 30 Higher genus; modular invariance etc Z X 2 D −Im(τ)(`2+m2) 1-loop partition function ∼ d τd ` e i i τ In string theory, modular invariance removes the UV region. i τ ! τ + 1; τ ! −1/τ UV This property has been assumed to hold also in the Ambitwistor string, which describes only -1/2 0 1/2 massless modes. But QFTs are UV divergent in general, so how can this be ? 8 / 30 Recap • Un- gauged-fixed version ambitwistor string ? • Where did the anti-holomorphic degrees of freedom go ? • High vs low energy limit of string theory ? • Higher loops ? • Modular invariance ? • etc. All of this motivates for looking or a fundamental origin of the ambitwistor string. 9 / 30 Tensionless (=Null) strings 9 / 30 Tensile strings Z 2 p Nambu-Goto action; SNG = −T d σ −g ; worldsheet string tension T ∼ 1/α0, σβ = (τ; σ) worldsheet coordinates, @X µ @X ν g = det g ; g = G αβ αβ @σα @σβ µν µ 0 with X (τ; σ) the coordinates of the string and @τ ≡ _ @σ ≡ How to take T ! 0? −! Hamiltonian formulation Z SH = P · X_ − H p @L @ −g with Pµ = = : the tension goes in H. @X_µ @X_µ 10 / 30 3 Integrate P out LST tensionless string action, Z α β SLST = V V @αX @βX : with V α a vector density. δS β δV α V @αX @βX = 0, so gαβ is null \tensionless ≡ null"4 Hamiltonian phase-space action ( P2 + T 2X 02 = 0 Find the two Virasoro constraints P · X 0 = 0 Hamiltonian = constraints (because of diffeo. invariance). Z Z 0 2 2 02 SH = P · X_ − H = P · (X_ − µX ) − λ(P + T X ) 3Lindstr¨om,B. Sundborg, and Theodoridis 1991. 4Schild 1977. 11 / 30 Hamiltonian phase-space action ( P2 + T 2X 02 = 0 Find the two Virasoro constraints P · X 0 = 0 Hamiltonian = constraints (because of diffeo. invariance). At T = 0, Z Z 0 2 SH = P · X_ − H = P · (X_ − µX ) − λP 3 Integrate P out LST tensionless string action, Z α β SLST = V V @αX @βX : with V α a vector density. δS β δV α V @αX @βX = 0, so gαβ is null \tensionless ≡ null"4 3Lindstr¨om,B. Sundborg, and Theodoridis 1991. 4Schild 1977. 11 / 30 R α 5 R Open question; what DV ? In string theory, Dhαβ gives the moduli space integration. Here it should give the Scattering equations at tree-level, and some precise first principle prescription at loop-level. Work in progress with E. Casali and Y. Herfray. 5Bo Sundborg 1994. 12 / 30 Nature of the Null Symmetry, with Casali & Herfray. Back to basics. R p ττ 2 Worldline action S = −gg (@τ x) . Diffeomorphism invariant. Go to first order Z e (p@ x − p2) t 2 Still have diffeos, τ ! τ + (τ) But seem to have new symmetry δx =x _ δx = αp δp =p _ δp = 0 δe = (e)_ δe =α _ These two symmetries should not be considered as different, they differ by an \equation of motion" symmetry.6 6See Henneaux & Teitelboim, Quantization of Gauge Systems 13 / 30 Nature of the Null Symmetry, with Casali & Herfray. There is a sense in whichz ¯ is the time on the ambitwistor/null string worldsheet, and such that the anti-holomorphic diffeos are transmuted to the null symmetry. Stay tuned ! Does this say something on KLT ? Maybe on KLT orthogonality ? 14 / 30 Classical analysis 14 / 30 Tensionless limit tensionless limit: wild fluctuation classically nearest neighboor interaction PX 0 vibrational energy 15 / 30 Identical to the ambitwistor string action Z SA = P · @¯X Remark: If integrate P out, get second order form Z 1 S = (@ X )2 λ − λ = 0 is a singular gauge. Ambitwistor gauge Casali and Tourkine 2016 Z S = P · (X_ − µX 0) − λP2 Gauge fix µ = 1; λ = 0 @± = @τ ± @σ. Keep σ ' σ + 2π periodicity. 16 / 30 Ambitwistor gauge Casali and Tourkine 2016 Z Z 0 2 S = P · (X_ − µX ) − λP = P · @−X Gauge fix µ = 1; λ = 0 @± = @τ ± @σ. Keep σ ' σ + 2π periodicity. Identical to the ambitwistor string action Z SA = P · @¯X Remark: If integrate P out, get second order form Z 1 S = (@ X )2 λ − λ = 0 is a singular gauge. 16 / 30 Constraint algebra Crucial part of our analysis in Casali and Tourkine 2016 was to show how the constraints match when you go to the gauge λ = 0. For what follows, just note that 0 L0 zero mode of P · X = angular momentum operator 2 M0 zero mode of P = mass operator 17 / 30 Quantization 17 / 30 Quantization: \f; g ! −i[ ; ]" Weyl ordering Normal ordering ( xν pµ if n > 0; : pµxν := xν pµ : pµxν := m n n m m n n m µ ν pn xm if m > 0; ( xnj0iA = 0 pnj0iHS = 0; 8n 2 Z 8n > 0 pnj0iA = 0 • No critical dimension • Critical dimension = 10 • Continuous mass spectrum • Only massless states of higher spins • Amplitudes ? • CFT techniques 18 / 30 8 L jphysi = 0; m > 0 <> m Spectrum: (L0 − 2)jphysi = 0; > :Mmjphysi = 0; m ≥ 0 µ ν µ [I J] =) jphysi 2 p−1p−1j0i; p−1p−1µj0i; p−1y−1j0i Bosonic ambitwistor/null string 8 µ [L ; L ] = (n − m)L + d m(m2 − 1)δ ; d = η <> n m n+m 6 m+n µ [Ln; Mm] = (n − m)Mn+m ; > :[Mn; Mm] = 0 ; P Normal ordering ambiguity; L0 = nxnp−n 19 / 30 8 L jphysi = 0; m > 0 <> m Spectrum: (L0 − 2)jphysi = 0; > :Mmjphysi = 0; m ≥ 0 µ ν µ [I J] =) jphysi 2 p−1p−1j0i; p−1p−1µj0i; p−1y−1j0i Bosonic ambitwistor/null string 8 µ [L ; L ] = (n − m)L + d m(m2 − 1)δ ; d = η <> n m n+m 6 m+n µ [Ln; Mm] = (n − m)Mn+m ; > :[Mn; Mm] = 0 ; P Normal ordering ambiguity; L0 = n : xnp−n : 19 / 30 Bosonic ambitwistor/null string 8 µ [L ; L ] = (n − m)L + d m(m2 − 1)δ ; d = η <> n m n+m 6 m+n µ [Ln; Mm] = (n − m)Mn+m ; > :[Mn; Mm] = 0 ; Normal ordering ambiguity; L0 ! L0 − 2 8 L jphysi = 0; m > 0 <> m Spectrum: (L0 − 2)jphysi = 0; > :Mmjphysi = 0; m ≥ 0 µ ν µ [I J] =) jphysi 2 p−1p−1j0i; p−1p−1µj0i; p−1y−1j0i 19 / 30 What goes wrong in the bosonic/heterotic versions Ambitwistor strings in curved space: Adamo, Casali, and Skinner 2015 argued that they are not spacetime diffeo invariant. Killer reason. Spectrum reason ? You actually have other states that are eigenstates of L0 − 2, but that are not mass eigenstates: y−2j0i and p−2j0i. You can try to diagonalize the basis but then you end up with wrong sign commutation relations which seem to indicate breakdown of unitarity.