On the Null origin of the Ambitwistor String

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

Cachazo, He, and Yuan 2014: breakthrough! n dz i δ ( sc. eqns. ) F (k i, ɛ j, z i ) i=4 z i k i (k 2 i = 0) Scattering Equations: degree (n 3)! polynomial eqs. Theory-dependent part. ɛ i, k j = polarisations, momenta 4 / 30

Ambitwistor string Mason and Skinner 2014 Chiral (holomorphic) string living in Ambitwistor space. S = P X X P X P 2 = 0 Spectrum: type II SUGRA in d = 10 (there is no α at all). Same as low energy spectrum of string theory. Scattering equations from P 2 = 0 Extend to higher genus amplitudes, 1 + +... 1 Adamo, Casali, and Skinner 2014; Adamo and Casali 2015. 5 / 30

Scattering equations & Gross-Mende P 2 = 0 j k i k j z i z j = 0 Fairlie and Roberts 1972 Govern the saddle point of the the Gross-Mende α of string theory. Why do the scattering equations have to do with both α 0 and α? 6 / 30

Loop scattering equations on the sphere Our sphere prescription succeeded at one and two loops 2. field theory propagators Where do these formulae come from really? How to generalize them? 2 Geyer, Mason, Monteiro, and Tourkine 2015; Geyer, Mason, Monteiro, and Tourkine 2016a; Geyer, Mason, Monteiro, and Tourkine 2016b. 7 / 30

Higher genus; modular invariance etc 1-loop partition function d 2 τd D l e Im(τ)(l2 +mi 2) i τ Field theory integration domain i UV -1/2 0 1/2 8 / 30

Higher genus; modular invariance etc 1-loop partition function d 2 τd D l e Im(τ)(l2 +mi 2) i τ In string theory, modular invariance removes the UV region. i -1/2 0 1/2 UV τ τ + 1, τ 1/τ This property has been assumed to hold also in the Ambitwistor string, which describes only massless modes. 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 Nambu-Goto action; S NG = T worldsheet d 2 σ g, string tension T 1/α, σ β = (τ, σ) worldsheet coordinates, g = det g αβ, g αβ = X µ σ α X ν σ β G µν with X µ (τ, σ) the coordinates of the string and τ σ How to take T 0? Hamiltonian formulation ( ) S H = P Ẋ H with P µ = L = g : the tension goes in H. Ẋ µ Ẋ µ 10 / 30

Hamiltonian phase-space action Find the two Virasoro constraints { P 2 + T 2 X 2 = 0 P X = 0 Hamiltonian = constraints (because of diffeo. invariance). ( ) ( ) S H = P Ẋ H = P (Ẋ µx ) λ(p 2 + T 2 X 2 ) 3 Lindström, B. Sundborg, and Theodoridis 1991. 4 Schild 1977. 11 / 30

Hamiltonian phase-space action Find the two Virasoro constraints { P 2 + T 2 X 2 = 0 P X = 0 Hamiltonian = constraints (because of diffeo. invariance). At T = 0, ( ) ( S H = P Ẋ H = P (Ẋ µx ) λp 2) Integrate P out LST tensionless string action, 3 S LST = V α V β α X β X. with V α a vector density. δs δv α V β α X β X = 0, so g αβ is null 3 Lindström, B. Sundborg, and Theodoridis 1991. 4 Schild 1977. tensionless null 4 11 / 30

Open question; what DV α? 5 In string theory, Dh αβ 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. 5 Bo Sundborg 1994. 12 / 30

Nature of the Null Symmetry, with Casali & Herfray. Back to basics. Worldline action S = gg ττ ( τ x) 2. Diffeomorphism invariant. Go to first order (p t x e 2 p2 ) Still have diffeos, τ τ + ɛ(τ) But seem to have new symmetry δx = ẋɛ δp = ṗɛ δe = (ɛe) δx = αp δp = 0 δe = α These two symmetries should not be considered as different, they differ by an equation of motion symmetry. 6 6 See Henneaux & Teitelboim, Quantization of Gauge Systems 13 / 30

Nature of the Null Symmetry, with Casali & Herfray. There is a sense in which z 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 vibrational energy 15 / 30

Ambitwistor gauge Casali and Tourkine 2016 S = P (Ẋ µx ) λp 2 Gauge fix µ = 1, λ = 0 ± = τ ± σ. Keep σ σ + 2π periodicity. 16 / 30

Ambitwistor gauge Casali and Tourkine 2016 S = P (Ẋ µx ) λp 2 = P X Gauge fix µ = 1, λ = 0 ± = τ ± σ. Keep σ σ + 2π periodicity. Identical to the ambitwistor string action S A = P X Remark: If integrate P out, get second order form 1 S = λ ( X ) 2 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 L 0 zero mode of P X = angular momentum operator M 0 zero mode of P 2 = mass operator 17 / 30

Quantization 17 / 30

Quantization: {, } i[, ] Weyl ordering Normal ordering { : p n µ xm ν := xmp ν n µ : p n µ xm ν x ν := mp n µ if n > 0, p n µ xm ν if m > 0, { x n 0 A = 0 p n 0 HS = 0, n Z n > 0 p n 0 A = 0 No critical dimension Critical dimension = 10 Continuous mass spectrum Only massless states of higher spins Amplitudes? CFT techniques 18 / 30

Bosonic ambitwistor/null string [L n, L m ] = (n m)l n+m + d 6 m(m2 1)δ m+n, d = η µ µ [L n, M m ] = (n m)m n+m, [M n, M m ] = 0, Normal ordering ambiguity; L 0 = nx n p n 19 / 30

Bosonic ambitwistor/null string [L n, L m ] = (n m)l n+m + d 6 m(m2 1)δ m+n, d = η µ µ [L n, M m ] = (n m)m n+m, [M n, M m ] = 0, Normal ordering ambiguity; L 0 = n : x n p n : 19 / 30

Bosonic ambitwistor/null string [L n, L m ] = (n m)l n+m + d 6 m(m2 1)δ m+n, d = η µ µ [L n, M m ] = (n m)m n+m, [M n, M m ] = 0, Normal ordering ambiguity; L 0 L 0 2 Spectrum: L m phys = 0, m > 0 (L 0 2) phys = 0, M m phys = 0, m 0 = phys p µ 1 pν 1 0, p µ 1 p 1µ 0, p[i 1 y J] 1 0 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 L 0 2, but that are not mass eigenstates: y 2 0 and p 2 0. You can try to diagonalize the basis but then you end up with wrong sign commutation relations which seem to indicate breakdown of unitarity. 20 / 30

Spinning ambitwistor string Add a pair of real fermions Ψ a, r = 1, 2, with new constraints Spectrum; P 2 = 0 P X + i 2 Ψ a Ψ a = 0 a=1,2 Ψ a P = 0 (a = 1, 2) Neveu-Schwarz: p 1 0, ψ 1/2 a ψb 1/2 0, truncated down to ψ 1/2 1 ψ2 1/2 0 after GSO-like projection. Similar for Ramond, this is then 10-d type II supergravity 21 / 30

Relation to tensionful strings 21 / 30

Left and right movers X (σ, τ) = x 0 + pτ + 1 n (α ne in(σ τ) + α n e in(σ+τ) ) 22 / 30

Left and right movers X (σ, τ) = x 0 + pτ + 1 n (α ne in(σ τ) + α n e in(σ+τ) ) { αn = 1 2 T p n in T x n α n = 1 2 T p n in T x n, [α n, α m ] = [ α n, α m ] = nδ m+n,0 In terms of these modes, the ambitwistor vacuum obeys α n 0 A = α n 0 A = 0, n > 0, in contrast with the string theory vacuum, α n 0 S = α n 0 S = 0, n > 0 22 / 30

Left and right movers L n, L n Virasoro generators L n = (L n L n ) = k : p n k x k : 2δ n,0 M n = T (L n + L n ) = k p n k p k + 2T 2 (k n)kx n k x k. Normal ordering constant; in string theory, both L 0, L 0 have 1. Due to the twisted ordering, L 0 switches to +1 and L 0 gets a 2. The constant is transfered from the mass operator L 0 + L 0 to the angular momentum (level matching) 7 L 0 L 0 2 7 Hwang, Marnelius, and Saltsidis 1999; W. Siegel 2015. 23 / 30

Tensile deformation of the ambitwistor string Hwang, Marnelius, and Saltsidis 1999, Hohm, Warren Siegel, and Zwiebach 2014, W. Siegel 2015; Huang, Warren Siegel, and Yuan 2016; Leite and Warren Siegel 2016 P 2 P 2 + T 2 ( X ) 2. The vertex ops for graviton, b-field and dilaton get some T dependence. And get two massive spin two states V = c cɛ µν (P µ P ν + T 2 X µ X ν ± P (µ X ν) )e ik X with masses k 2 = 4T 2. Same as [HSZ]. These are ghosts. 8 They mix up with the gravity when you send T 0. 8 Leite and Warren Siegel 2016. 24 / 30

Connection with Gross & Mende? In both the Ambitwistor str and Gross & Mende, the constraint P 2 + T 2 X 2 descends to P 2. In the ambitwistor string, the constraint imposes a localisation onto the scattering equations via BRST localisation. In string theory, the localisation on the Virasoro constraints T ±,± = (P ± TX ) 2 imposes the reduction of the path integral down to the moduli space. 25 / 30

What about the Higher-Spin theory then? Probably very hard to compute amplitudes in the HS-like theory, and they should be anyway non-perturbative: String theory is exponentially soft at high energies, A exp ( α s ln s) exp ( 1T )... e 1/g(...) correction, typically non perturbative. 26 / 30

Perspectives Path integral over the V α s to determine the integration cycle at loop-level from first principles Understand the complexification 27 / 30

Recap 1. What I have not said: the ambitwistor string is the high energy (or α 0) limit of string theory 2. Classically, the ambitwistor string is obtained form a Null String action, which is a tensionless string. 3. Siegel and co s story is doing is equivalent to this story, Historical remark: the scattering equations etc. could have been discovered in the 80s! 28 / 30

The ambitwistor string is a null string Worldline Strings Null strings α 0 α GM SUGRA QM strings Higher-spin Ambitwistor string Thank you 29 / 30

References I Adamo, Tim and Eduardo Casali. Scattering Equations, Supergravity Integrands, and Pure Spinors. In: JHEP 05 (2015), p. 120. doi: 10.1007/JHEP05(2015)120. arxiv: 1502.06826 [hep-th]. Adamo, Tim, Eduardo Casali, and David Skinner. A Worldsheet Theory for Supergravity. In: JHEP 02 (2015), p. 116. doi: 10.1007/JHEP02(2015)116. arxiv: 1409.5656 [hep-th].. Perturbative Gravity at Null Infinity. In: Class. Quant. Grav. 31.22 (2014), p. 225008. doi: 10.1088/0264-9381/31/22/225008. arxiv: 1405.5122 [hep-th]. Cachazo, Freddy, Song He, and Ellis Ye Yuan. Scattering of Massless Particles in Arbitrary Dimensions. In: Phys.Rev.Lett. 113.17 (2014), p. 171601. doi: 10.1103/PhysRevLett.113.171601. arxiv: 1307.2199 [hep-th]. Casali, Eduardo and Piotr Tourkine. On the Null Origin of the Ambitwistor String. In: JHEP (2016). arxiv: 1606.05636. Fairlie, D. B. and D. E. Roberts. Dual Models without Tachyons - a New Approach. In: (1972). Geyer, Yvonne, Lionel Mason, Ricardo Monteiro, and Piotr Tourkine. Loop Integrands for Scattering Amplitudes from the Riemann Sphere. In: Phys. Rev. Lett. 115.12 (2015), p. 121603. doi: 10.1103/PhysRevLett.115.121603. arxiv: 1507.00321 [hep-th].. One-loop amplitudes on the Riemann sphere. In: JHEP 03 (2016), p. 114. doi: 10.1007/JHEP03(2016)114. arxiv: 1511.06315 [hep-th].. Two-Loop Scattering Amplitudes from the Riemann Sphere. In: (2016). arxiv: 1607.08887 [hep-th]. Hohm, Olaf, Warren Siegel, and Barton Zwiebach. Doubled α -geometry. In: JHEP 02 (2014), p. 065. doi: 10.1007/JHEP02(2014)065. arxiv: 1306.2970 [hep-th]. 29 / 30

References II Huang, Yu-tin, Warren Siegel, and Ellis Ye Yuan. Factorization of Chiral String Amplitudes. In: (2016). arxiv: 1603.02588 [hep-th]. Hwang, Stephen, Robert Marnelius, and Panagiotis Saltsidis. A General BRST Approach to String Theories with Zeta Function Regularizations. In: J. Math. Phys. 40 (1999), pp. 4639 4657. doi: 10.1063/1.532994. arxiv: hep-th/9804003 [hep-th]. Leite, Marcelo M. and Warren Siegel. Chiral Closed strings: Four massless states scattering amplitude. In: (2016). arxiv: 1610.02052 [hep-th]. Lindström, U., B. Sundborg, and G. Theodoridis. The Zero Tension Limit of the Superstring. In: Phys. Lett. B253 (1991), pp. 319 323. doi: 10.1016/0370-2693(91)91726-C. Mason, Lionel and David Skinner. Ambitwistor Strings and the Scattering Equations. In: JHEP 1407 (2014), p. 048. doi: 10.1007/JHEP07(2014)048. arxiv: 1311.2564 [hep-th]. Schild, Alfred. Classical Null Strings. In: Phys. Rev. D16 (1977), p. 1722. doi: 10.1103/PhysRevD.16.1722. Siegel, W. Amplitudes for Left-Handed Strings. In: (2015). arxiv: 1512.02569 [hep-th]. Sundborg, Bo. Strongly Topological Interactions of Tensionless Strings. In: (1994). arxiv: hep-th/9405195 [hep-th]. 30 / 30