Ambitwistor strings, the scattering equations, tree formulae and beyond

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1 Ambitwistor strings, the scattering equations, tree formulae and beyond Lionel Mason The Mathematical Institute, Oxford Les Houches 18/6/2014 With David Skinner. arxiv: and Yvonne Geyer & Arthur Lipstein [Cf. also Cachazo, He, Yuan arxiv: , , , ]

2 String formulae for tree-level scattering Remarkable compact formulae: Twistor-string for N = 4 Yang-Mills [Witten, Roiban, Spradlin, Volovich, 2003/4]. Some (conformal-) supergravity amplitudes [Adamo, Mason 2012] N = 8 supergravity [Cachazo-Geyer, Cachazo-Skinner, CMS, 2012], ABJM [Huang,Lee 20012]. Pfaffian/current-correlator formulae for spins 2, 1, 0 in all dimensions [Cachazo, He, Yuan, 2013]. CS N=8 formula arises from N = 8 twistor-string [Skinner, 2013] This talk: Such formulae ambitwistor string theories : Ambitwistors A = complexified space of null geodesics. A = reduction of complex cotangent bdle of space-time. Theory is critical in 10d. In 4d, A = cotangent bdle of twistor space or dual twistors or new ambidextrous formulation. New family of infinite tension (α = 0) chiral analogues of standard strings that extend original twistor-string.

3 The scattering equations Take n null momenta k i R d, i = 1,..., n, k 2 i = 0, i k i = 0, define P : CP 1 C d P(σ) := n i=1 k i σ σ i, σ, σ i CP 1. Solve for σ i CP 1 with the scattering equations Res σi P(σ) P(σ) = k i P(σ i ) = n j=1 k i k j σ i σ j = 0. Then P(σ) P(σ) = 0 σ. For Mobius invariance P C d K, K = Ω 1,0 CP 1 only n 3 scattering equations are independent. There are (n 3)! solutions. First arose for high energy string scattering [Gross-Mende 1988]. Underpin twistor-string formulae also [Witten 2004].

4 The Cachazo-He-Yuan formulae Formulae for gravity, Yang-Mills and scalar amplitudes. Scatter n spin s massless particles, momenta k i, k 2 i = 0, polarizations ɛ 1i for spin 1, ɛ 1i ɛ 2i for spin-2 k i ɛ ri = 0, ɛ ri ɛ ri + α r k i, r = 1, 2. Introduce skew 2n 2n matrices ( ) A Cr M r =, A ij = k i k j σ i σ j,, C t r B rij = ɛ ri ɛ rj σ i σ j, B r C rij = k i ɛ rj σ i σ j, for i j and A ii = B ii = 0, but C rii = ɛ ri P(σ i ). Tree-level gravity ( amplitude ) in d-dims is sum M(1,..., n) = δ d Pf (M 1 )Pf (M 2 ) k i δ(k i P(σ i ))dσ i i CP 1n Vol SL(2, C) i [For YM, replace Pf (M 2 ) by Parke-Taylor ( i (σ i σ i 1 )) 1.]

5 Geometry of ambitwistor space Complexify real space-time M R M, and null covectors P. A := space of complex null geodesics with scale of P. A = T M P 2 =0 /{D 0} where D 0 := P = geodesic spray. D 0 has Hamiltonian P 2 wrt symplectic form ω = dp µ dx µ. Symplectic potential θ = P µ dx µ, ω = dθ, descend to A. Projectivise: PA := space of unscaled complex light rays. On PA, θ Ω 1 PA L is a holomorphic contact structure. Theorem (LeBrun 1983) The complex structure on PA determines M and conformal metric g. The correspondence is stable under arbitrary deformations of the complex structure of PA that preserve θ.

6 Linearized LeBrun correspondence Baston & M θ determines complex structure on PA via θ dθ d 2. So: Deformations of complex structure [δθ] H 1 (PA, L). Analyze with double fibration PT M P 2 =0 π 1 D 0 π 2 PA S M. For δg µν = e ik x ɛ µν on flat space-time δθ = δ(k P)e ik X ɛ µν P µ P ν. Delta-function support on k P = 0 the scattering equations. Proof: The correspondence between δg and δθ is π1 δθ = j on PT M P 2 =0 for some j modulo P V (x). D0 j = D 0 π1 δθ = 0 so D 0j is holomorphic in (P, X), thus D 0 j = δg µν (X)P µ P ν for some variation in the metric δg. For momentum eigenstate, j = e ik X ɛ µνp µ P ν k P.

7 From null geodesics to chiral strings For real space-time (M R, g R ) dimension d, Phase space action: null geodesic γ, (X, P) : R T M R S = (P dx ep 2 /2), γ e Ω 1 (γ) is einbein and Lagrange multiplier for P 2 = 0. Gauge freedom δ(x, P, e) = (αp, 0, 2dα). Phase space of real null geodesics: A R := T M R P 2 =0 /{gauge} Complexify: γ Σ, Riemann surface, and (M R, g R ) (M, g). Ambitwistor string action: X : Σ M, P K X T M S = (P X e P 2 /2). with e Ω 0,1 T, where K = Ω 1,0 Σ and T = T 1,0 Σ. e again enforces P 2 = 0, flat space gauge freedom: δ(x, P, e) = (αp, 0, 2 α). Ambitwistor space: A = T M P 2 =0 /{gauge}.

8 From null geodesics to chiral strings For real space-time (M R, g R ) dimension d, Phase space action: null geodesic γ, (X, P) : R T M R S = (P dx ep 2 /2), γ e Ω 1 (γ) is einbein and Lagrange multiplier for P 2 = 0. Gauge freedom δ(x, P, e) = (αp, 0, 2dα). Phase space of real null geodesics: A R := T M R P 2 =0 /{gauge} Complexify: γ Σ, Riemann surface, and (M R, g R ) (M, g). Ambitwistor string action: X : Σ M, P K X T M S = (P X e P 2 /2). with e Ω 0,1 T, where K = Ω 1,0 Σ and T = T 1,0 Σ. e again enforces P 2 = 0, flat space gauge freedom: δ(x, P, e) = (αp, 0, 2 α). Ambitwistor space: A = T M P 2 =0 /{gauge}.

9 Quantizing spin 0 ambitwistor string To quantize, gauge fix S = (P X e P 2 /2). with e = 0 and ghosts ( b, c) (K 2, T ) plus usual (b, c) (K 2, T ) for diffeos S ghost = b c + b c. This gives BRST operator Q = ct + cp 2. We have central charge C = 2d so to quantize consistently Q 2 = 0 d = 26.

10 Vertex operators and amplitudes Integrated vertex ops = perturbations of action δg. Action is θ = P X so integrated vertex operator is V i = δθ(σ i ) = δ(k i P(σ i )) e ik X(σ i ) ɛ iµν P µ (σ i )P ν (σ i ). Σ Σ Quantum consistency implies field equations: {Q, V i } = 0 k 2 = 0, k µ ɛ µν = 0. Fixed vertex operators provide Fadeev Popov determinants for fixing remaining gauge symmetries G = SL(2, C) C 3 for Mobius on CP 1 and translations along D 0. Replace fixed vertex ops by quotient by G to give amplitude as path-integral D[X, P,...] n M(1,..., n) = e is V i. Vol G i=1

11 Evaluation of amplitude Take e ik i X(σ i ) factors into action to give S = 1 P X + 2π ik X(σ i ). 2π Σ i Gives field equations X = 0, P = 2π i ikδ2 (σ σ i ). Solutions X(σ) = X = const., P(σ) = i k i σ σ i dσ. Thus path-integral reduces to M(1,..., n) = d d X e i n δ(k j k j X i=1 i P) ɛ iµν P µ (σ i )P ν (σ i ) R d (CP 1 ) n 3 Vol G = δ d ( k i ) δ(k i P) ɛ iµν P µ (σ i )P ν (σ i ) i (CP 1 ) n 3 i We see P(σ) appearing and scattering equations. Unfortunately: no good interpretation of these as amplitudes.

12 Evaluation of amplitude Take e ik i X(σ i ) factors into action to give S = 1 P X + 2π ik X(σ i ). 2π Σ i Gives field equations X = 0, P = 2π i ikδ2 (σ σ i ). Solutions X(σ) = X = const., P(σ) = i k i σ σ i dσ. Thus path-integral reduces to M(1,..., n) = d d X e i n δ(k j k j X i=1 i P) ɛ iµν P µ (σ i )P ν (σ i ) R d (CP 1 ) n 3 Vol G = δ d ( k i ) δ(k i P) ɛ iµν P µ (σ i )P ν (σ i ) i (CP 1 ) n 3 i We see P(σ) appearing and scattering equations. Unfortunately: no good interpretation of these as amplitudes.

13 Spinning light rays and super ambitwistor space To get Pfaffians include RNS spin vectors Ψ µ r, fermions: S[X, P, Ψ] = P µ dx µ e 2 2 P µp µ + g µν Ψ µ r dψ ν r χ r P µ Ψ µ r χ r constraints P Ψ r = 0 that generate worldline N = 2 susy D r = Ψ r r=1 X + P, {D r, D s } = δ rs D 0. Ψ r Super ambitwistor space: A s = symplectic quotient of (X, P, Ψ r )-space by P 2, P Ψ r. Symplectic potential: θ = P dx + Ψ r dψ r Super LeBrun correspondence holds with perturbations 2 δθ = e ik X δ(k P) ɛ rµ (P µ + Ψ µ r k Ψ r ). r=1 Note: polarization states ɛ 1µ ɛ 2ν NS sector of type II sugra.

14 Super ambitwistor strings Use chiral RNS-like action S[X, P, Ψ] = Σ P X e 2 P µp µ + with N = 2 susy (degenerate). 2 Ψ r Ψ r + χ r P Ψ r r=1 To quantize, gauge fix χ r = 0 bosonic ghosts (β r, γ r ) in (K 3/2, T 1/2 ) for fermionic symmetry (and (b, c), ( b, c)). We obtain BRST operator Q = ct + cp 2 + γ r P Ψ r. For Q 2 = 0 central charge C must vanish C = 2d + d 2 + d = 3(D 10) So critical in d = 10 dimensions.

15 Amplitudes Integrated vertex operator V i = e ik X(σ i ) δ(k P(σ i )) Σ 2 ɛ rµ (P µ (σ i ) + Ψ µ r (σ i )k Ψ r (σ i )) r=1 need two fixed operators for γ r zero modes (fixing susy) U i = e k i X(σ i ) r ɛ r Ψ r (σ i ) and an extra fixed one to fix 3rd c and c zero modes 2 V i = ɛ rµ (P µ (σ i ) + Ψ µ r (σ i )k Ψ r (σ i )) r=1 So amplitudes are given by M(1,..., n) = c 1 c 1 γ r1 U 1 c 2 c 2 γ r2 U 2 c 3 c 3 V 3 V 4... V n. r r Much works as before giving δ d ( i k i) i δ(k i P) etc..

16 Amplitude formulae with Pfaffians New ingredient is the correlation function of the Ψ s. Ψ 1, Ψ 2 independent so contractions computed separately. Ψ s appear twice in V i, as k i Ψ i or ɛ i Ψ i. Contractions give for example A ij := k i Ψ i k j Ψ j = k i k j σ i σ j, B ij := ɛ i Ψ i ɛ j Ψ j = ɛ i ɛ j σ i σ j. For C ij = k i ψ i ɛ j ψ j, P(σ i ) ɛ i gives diagonal entry. Two k Ψs are missing in U i + ghost contribution Pf (M). Theorem We obtain CHY formula M(1,..., n) = δ d ( i k i ) CP 1n Pf (M 1 )Pf (M 2 ) Vol SL(2, C) δ(k i P(σ i ))dσ i i

17 Other models We can start with other formulations of null superparticles Heterotic model: as above but r = 1 and current algebra (SO(32) or E 8 E 8 for Q 2 = 0) CHY Yang-Mills formula. Bosonic case +2 current algebras CHY scalar formula. Green-Schwarz version: S = P X + P µ γ µ αβ θα θ β. Berkovits has pure spinor version S = P X + p α θ α +....

18 Summary We have chiral α = 0 ambitwistor strings based on LeBrun s correspondence that gives theory underlying CHY formulae NS sector of type II sugra extended to Ramond as in RNS string via spin-operator from bosonizing Ψs. Incorporates KLT/BCJ ideas. A new representation for the loop integrand?[adamo, Casali, Skinner] At genus g, P is a 1-form and acquires dg zero-modes. These are the loop momenta for g-loops. E.g., at 1-loop get sum over spin structures of ( M (1) n (α; β) = δ 10 i n j=2 k i ) d 10 p dτ δ ( ) P 2 (σ 1 ; τ) dσ j δ(k j P(σ j ))) ϑ α(τ) 4 ϑ β (τ) 4 η(τ) 24 Pf(M α ) Pf( M β ) But how to get rid of ϑ-functions?

19 Original 4d ambitwistor space Ambitwistors in 4 dimensions with Yvonne Geyer & Arthur Lipstein A = {(Z, W ) PT PT Z W = 0}/{Z Z W W }. Here T = C 4 N, N 3 for Yang-Mlls, N 7 for gravity. Z = (λ α, µ α, χ a ), W = ( λ α, µ α ), α = 0, 1, α = 0, 1, a = 1,... N. Relates to previous version by P α α = λ α λ α, µ α = ix α α λ α, µ α = ix α α λ α. Symplectic potential Θ = i (W dz Z dw ) = P dx 2

20 Ambitwistor string in 4 dims The symplectic potential Θ leads to action S = W Z Z W + az W. Σ (cf., Berkovits twistor-string, RSV etc.). But: now ambidextrous so take Z, W K 1/2 not arbitrary. Use wave fns from both PT and PT YM vertex ops V i = Ṽ i = dsi δ 2 (λ i s i λ(σ i ))e is i [µ λ i ] J t i s i dsi δ 2 ( λ i s i λ(σ i ))e is i µ λ i J t i s i

21 New 4d formulae N k MHV amplitude M (1,..., n) = Ṽ1... ṼkV k+1... V n. Same strategy as before with σ α = 1 s (1, σ) gives λ(σ) = k i=1 λ i n (σ σ i ), λ(σ) = For Yang-Mills obtain amplitude A(1,..., n) = n i=1 d 2 σ i (σ i σ i+1 ) i=k+1 k δ 2 ( λ i λ(σ)) i=1 λ i (σ σ i ). n i=k+1 δ 2 (λ i λ(σ i )) Gravity: adapt Skinner model, replace 1 (i i+1) by det H H ij = H ij = i j (i j), i, j k, [i j] Hij = (i j), i, j > k, ( ) H 0, H 0 H ii = H il. l i

22 Summary for 4d models Formulae proved by comparison to original twistor strings. Comparison to CHY: det H = Pf (M). Valid for any degree of SUSY unlike original twistor-strings. Noncritical/anomalous as they stand, but perhaps not with additional matter and then unobstructed at loops. String field theory action should encode colour-kinematics...

23 Thank You The end