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 → ∞ of string theory.
Why do the scattering equations have to do with both α0 → 0 and α0 → ∞ ?
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 √ 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 √ ∂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 √ 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: “{, } → −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, ( xn|0iA = 0 pn|0iHS = 0, ∀n ∈ Z ∀n > 0 pn|0iA = 0
• No critical dimension • Critical dimension = 10 • Continuous mass spectrum • Only massless states of higher spins • Amplitudes ? • CFT techniques
18 / 30 L |physi = 0, m > 0 m Spectrum: (L0 − 2)|physi = 0, Mm|physi = 0, m ≥ 0
µ ν µ [I J] =⇒ |physi ∈ p−1p−1|0i, p−1p−1µ|0i, p−1y−1|0i
Bosonic ambitwistor/null string
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 L |physi = 0, m > 0 m Spectrum: (L0 − 2)|physi = 0, Mm|physi = 0, m ≥ 0
µ ν µ [I J] =⇒ |physi ∈ p−1p−1|0i, p−1p−1µ|0i, p−1y−1|0i
Bosonic ambitwistor/null string
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
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
L |physi = 0, m > 0 m Spectrum: (L0 − 2)|physi = 0, Mm|physi = 0, m ≥ 0
µ ν µ [I J] =⇒ |physi ∈ p−1p−1|0i, p−1p−1µ|0i, p−1y−1|0i
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−2|0i and p−2|0i. 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 Ψa, r = 1, 2, with new constraints
P2 = 0 0 i X a 0a P · X + 2 Ψ · Ψ = 0 a=1,2 Ψa · P = 0 (a = 1, 2)
Spectrum; a b Neveu-Schwarz: p−1|0i, ψ−1/2ψ−1/2|0i, truncated down to 1 2 ψ−1/2ψ−1/2|0i after GSO-like projection. Similar for Ramond, this is then 10-d type II supergravity
21 / 30 Relation to tensionful strings
21 / 30 ( √ α = √1 p − in T x n 2 T n n √ , [αn, αm] = [˜αn, α˜m] = nδm+n,0 α˜ = √1 p − in T x n 2 T −n −n
In terms of these modes, the ambitwistor vacuum obeys
αn|0iA =α ˜−n|0iA = 0 , ∀ n > 0 ,
in contrast with the string theory vacuum,
αn|0iS =α ˜n|0iS = 0 , ∀ n > 0
Left and right movers
X 1 X (σ, τ) = x + pτ + (α ein(σ−τ) +α ˜ ein(σ+τ)) 0 n n n
22 / 30 Left and right movers
X 1 X (σ, τ) = x + pτ + (α ein(σ−τ) +α ˜ ein(σ+τ)) 0 n n n ( √ α = √1 p − in T x n 2 T n n √ , [αn, αm] = [˜αn, α˜m] = nδm+n,0 α˜ = √1 p − in T x n 2 T −n −n
In terms of these modes, the ambitwistor vacuum obeys
αn|0iA =α ˜−n|0iA = 0 , ∀ n > 0 ,
in contrast with the string theory vacuum,
αn|0iS =α ˜n|0iS = 0 , ∀ n > 0
22 / 30 Left and right movers
˜ Ln, Ln Virasoro generators X Ln = (Ln − L˜−n) = k : pn−k · xk : −2δn,0 X 2 Mn = T (Ln + L˜−n) = pn−k · pk + 2T (k − n)kxn−k · xk . k
Normal ordering constant; in string theory, both L0, L˜0 have −1. Due to the twisted ordering, L˜0 switches to +1 and L0 gets a −2.
The constant is transfered from the mass operator L0 + L˜0 to the angular momentum (level matching)7
L0 − L˜0 − 2
7Hwang, 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
P2 → P2 + T 2(∂X )2.
The vertex ops’ for graviton, b-field and dilaton get some T dependence. And get two massive spin two states
µ ν 2 µ ν (µ ν) ik·X V = cc˜µν(P P + T ∂X ∂X ± P ∂X )e
with masses k2 = ∓4T 2. Same as [HSZ]. These are ghosts.8 They mix up with the gravity when you send T → 0.
8Leite and Warren Siegel 2016. 24 / 30 Connection with Gross & Mende ?
In both the Ambitwistor str and Gross & Mende, the constraint P2 + T 2X 02 descends to P2. 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 0 2 T±,± = (P ± TX ) 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,
1 A ∼ exp (−α0s ln s) ∼ exp − ... T −1/g(...) e 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 α0 → 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
0 Worldline Strings α →∞ Null strings α0→0
~ GM Ambitwistor SUGRA QM strings Higher-spin string ×
Thank you
29 / 30 ReferencesI
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 α0-geometry”. In: JHEP 02 (2014), p. 065. doi: 10.1007/JHEP02(2014)065. arXiv: 1306.2970 [hep-th].
29 / 30 ReferencesII
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].
Lindstr¨om,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