On the Null origin of the Ambitwistor String

Transcription

1 On the Null origin of the Ambitwistor String Piotr Tourkine, University of Cambridge, DAMTP Dec. 6th, 2016, QCD meets Gravity, Bhaumik institute

2 Based on: Eduardo Casali and Piotr Tourkine. On the Null Origin of the Ambitwistor String. In: JHEP (2016). arxiv: and work in progress with Eduardo Casali and Yannick Herfray. 2 / 30

3 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

4 Motivations 3 / 30

5 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

6 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 Magic: solutions F = n-point field theory amplitude Jacobian (gravity, Yang-Mills, scalar, etc) 4 / 30

7 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, Adamo, Casali, and Skinner 2014; Adamo and Casali / 30

8 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

9 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

10 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

11 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

12 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. But QFTs are UV divergent in general, so how can this be? 8 / 30

13 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

14 Tensionless (=Null) strings 9 / 30

15 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

16 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 Schild / 30

17 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 Schild tensionless null 4 11 / 30

18 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 / 30

19 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

20 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

21 Classical analysis 14 / 30

22 Tensionless limit tensionless limit: wild fluctuation classically nearest neighboor interaction PX vibrational energy 15 / 30

23 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

24 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

25 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

26 Quantization 17 / 30

27 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

28 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

29 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

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] / 30

31 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

32 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

33 Relation to tensionful strings 21 / 30

34 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

35 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

36 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 Hwang, Marnelius, and Saltsidis 1999; W. Siegel / 30

37 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 / 30

38 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

39 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

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

41 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

42 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

43 References I Adamo, Tim and Eduardo Casali. Scattering Equations, Supergravity Integrands, and Pure Spinors. In: JHEP 05 (2015), p doi: /JHEP05(2015)120. arxiv: [hep-th]. Adamo, Tim, Eduardo Casali, and David Skinner. A Worldsheet Theory for Supergravity. In: JHEP 02 (2015), p doi: /JHEP02(2015)116. arxiv: [hep-th].. Perturbative Gravity at Null Infinity. In: Class. Quant. Grav (2014), p doi: / /31/22/ arxiv: [hep-th]. Cachazo, Freddy, Song He, and Ellis Ye Yuan. Scattering of Massless Particles in Arbitrary Dimensions. In: Phys.Rev.Lett (2014), p doi: /PhysRevLett arxiv: [hep-th]. Casali, Eduardo and Piotr Tourkine. On the Null Origin of the Ambitwistor String. In: JHEP (2016). arxiv: 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 (2015), p doi: /PhysRevLett arxiv: [hep-th].. One-loop amplitudes on the Riemann sphere. In: JHEP 03 (2016), p doi: /JHEP03(2016)114. arxiv: [hep-th].. Two-Loop Scattering Amplitudes from the Riemann Sphere. In: (2016). arxiv: [hep-th]. Hohm, Olaf, Warren Siegel, and Barton Zwiebach. Doubled α -geometry. In: JHEP 02 (2014), p doi: /JHEP02(2014)065. arxiv: [hep-th]. 29 / 30

44 References II Huang, Yu-tin, Warren Siegel, and Ellis Ye Yuan. Factorization of Chiral String Amplitudes. In: (2016). arxiv: [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 doi: / arxiv: hep-th/ [hep-th]. Leite, Marcelo M. and Warren Siegel. Chiral Closed strings: Four massless states scattering amplitude. In: (2016). arxiv: [hep-th]. Lindström, U., B. Sundborg, and G. Theodoridis. The Zero Tension Limit of the Superstring. In: Phys. Lett. B253 (1991), pp doi: / (91)91726-C. Mason, Lionel and David Skinner. Ambitwistor Strings and the Scattering Equations. In: JHEP 1407 (2014), p doi: /JHEP07(2014)048. arxiv: [hep-th]. Schild, Alfred. Classical Null Strings. In: Phys. Rev. D16 (1977), p doi: /PhysRevD Siegel, W. Amplitudes for Left-Handed Strings. In: (2015). arxiv: [hep-th]. Sundborg, Bo. Strongly Topological Interactions of Tensionless Strings. In: (1994). arxiv: hep-th/ [hep-th]. 30 / 30