Ambitwistor Strings for Four Dimensions Yvonne Geyer Mathematical - PowerPoint PPT Presentation

Ambitwistor Strings for Four Dimensions Yvonne Geyer Mathematical Institute, Oxford New Geometric Structures in Scattering Amplitudes September 23, 2014 - Oxford Based on YG, Arthur Lipstein and Lionel Mason arXiv: 1404.6219, 1406.1462 Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 1 / 38

Motivation Motivation Since the formulation of the twistor string theories [Witten, Berkovits, Skinner] , many remarkable formulae for tree-level scattering amplitudes have been developed [RSVW, ACCK], [Hodges, Cachazo-YG, Cachazo-Skinner, Cachazo-He-Yuan] . This inevitably raises questions regarding the underlying theories: What is the origin of these representations of Yang-Mills and gravity scattering amplitudes? Recent work has focussed on answering this question, and beyond providing a geometric explanation of the formulae, it also facilitated extensions in various directions. In particular, the CHY [Cachazo-He-Yuan] representation has been understood as arising from string theories in ambitwistor space, the space of null geodesics. Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 2 / 38

Motivation Scattering equations and CHY formulae Scattering equations and CHY formulae [Cachazo-He-Yuan] P ( σ ) holomorphic map from Riemann sphere into momentum space, P ( σ ) = � P : CP 1 → CP d , k j σ − σ j . j Scattering equations � k i · k j k i · P ( σ i ) = = 0 σ i − σ j j � i Representation of YM and gravity scattering amplitudes � � n � ′ i = 1 d σ i i ¯ 1 δ ( k i · P ( σ i )) Pf ′ (Ψ) A = � n Vol SL ( 2 ; C ) i = 1 σ i , i + 1 � � n � ′ i = 1 d σ i i ¯ δ ( k i · P ( σ i )) Pf ′ (Ψ) Pf ′ (˜ M = Ψ) Vol SL ( 2 ; C ) Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 3 / 38

Motivation The Ambitwistor String in d=10 The Ambitwistor String in d=10 Ambitwistor space A Ambitwistor space = space of complex null geodesics in M C Symplectic quotient of cotangent bundle of (supersymmetric) spacetime ( X , P , Ψ) ∈ T ∗ M by constraints P 2 = 0 and Ψ r · P = 0 � � � � � � P 2 = 0 , Ψ r · P = 0 ( X µ , P µ , Ψ µ r ) ∈ T ∗ M A := {D 0 , D r } with Hamiltonian vector fields D 0 = P · ∇ , D r = Ψ r · ∇ + P · ∂ Ψ r A is a symplectic holomorphic manifold, with symplectic potential � ∂ X + 1 Θ = P · ¯ Ψ r · d Ψ r 2 r Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 4 / 38

Motivation The Ambitwistor String in d=10 The Ambitwistor String in d=10 RNS ambitwistor string [Mason-Skinner] (see also [Adamo-Casali-Skinner, Berkovits]) Complexify action of massless spinning particle � � 2 P 2 − χ r P · Ψ r P · ¯ r Ψ r · ¯ 1 ∂ X + 1 ∂ Ψ r − e S = 2 π 2 Geometrically, the action is obtained from the symplectic potential Θ , and the gauge fields e and χ r impose the constraints. This reduces the phase space to A . � 2 P 2 + � r γ r P · Ψ r + ˜ cT + ˜ c b BRST operator Q = 2 γ r γ r , nilpotent in d = 10 as in the usual superstring In particular, the correlation functions of appropriate VO can be shown to yield the CHY formulae. Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 5 / 38

Motivation Four dimensions Four dimensions In four dimensions, the space of null geodesics has an alternative spinorial representation in addition to the vector representation used in the formulation of the RNS ambitwistor string. This suggests that the ambitwistor string ideas can be implemented naturally to construct models for Yang-Mills and gravity. These models allow for any amount of supersymmetry, and the correlation functions lead to new, remarkably simple formulae for tree-level scattering amplitudes which are supported on the scattering equations parity invariant. Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 6 / 38

Motivation Four dimensions Outline Ambitwistor strings in d=4 1 Ambitwistor space Worldsheet Theory Yang-Mills 2 Gravity 3 Ambitwistor Strings at Null Infinity 4 Geometry and Symmetries Worldsheet theory Soft limits Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 7 / 38

Ambitwistor strings in d=4 Ambitwistor space Ambitwistor strings in d=4 Ambitwistor space A 4 Alternative twistorial representation: Z = ( λ α , µ ˙ α , χ r ) ∈ T = C 4 |N χ r ) ∈ T ∗ = C 4 |N µ α , ˜ W = (˜ α , ˜ λ ˙ Ambitwistor space is the quadric Z · W = 0 inside T × T ∗ � �� � � , ( Z I , W I ) ∈ T × T ∗ | Z · W = 0 A := Z ∂ ∂ Z − W ∂ ∂ W which can be seen as the symplectic quotient of T × T ∗ by the Hamiltonian Z · W . A is thus a symplectic manifold with the potential Θ = i 2 ( W · d Z − Z · d W ) Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 8 / 38

Ambitwistor strings in d=4 Ambitwistor space Ambitwistor strings in d=4 Ambitwistor space A 4 Comments: The incidence relations α = i ( x α ˙ α + i θ a α ˜ χ a = θ a α λ α µ ˙ θ ˙ α a ) λ α , µ α = − i ( x α ˙ α − i θ a α ˜ θ ˙ a )˜ χ a = ˜ θ ˙ a ˜ α α ˜ λ ˙ α , ˜ λ ˙ α realize a point in chiral Minkowski space as a quadric, CP 1 × CP 1 . α , then the null geodesic constraint P 2 = 0 α = λ α ˜ Define P α ˙ λ ˙ (appearing in the vectorial representation) is explicitly solved. Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 9 / 38

Ambitwistor strings in d=4 Worldsheet Theory Worldsheet Theory Motivation: In analogy to the ambitwistor string in d = 10, we will complexify the action of a massless spinning particle, the Ferber superparticle. Again, the action S is determined by the symplectic potential Θ , and the constraint Z · W = 0 is imposed by introducing a gauge field a ; � S = 1 W · ¯ ∂ Z − Z · ¯ ∂ W + a Z · W . 2 π Σ Here, ( Z , W ) are spinors on the worldsheet, ( Z , W ) ∈ Ω 0 (Σ , ( T × K 1 / 2 ) × ( T ∗ × K 1 / 2 )) Adding worldsheet gravity and gauge-fixing yields the BRST operator � Q = c ( W · ∂ Z − Z · ∂ W ) + uZ · W Note: In general anomalous! Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 10 / 38

Yang-Mills Vertex operators Yang-Mills Amplitudes Vertex Operators Introduce integrated and unintegrated vertex operators for self-dual and anti self-dual fields � � d s a δ 2 ( λ a − s a λ ) e is a ( [ µ ˜ λ a ]+ χ r ˜ η ar ) j · t a V ′ ¯ V ′ d σ a V ′ a = a = a s a � � d s a χ r η r � δ 2 (˜ ¯ λ a − s a ˜ λ ) e is a ( � ˜ µ λ a � +˜ a ) j · t a � d σ a � V a = V a = V a s a where j denotes a current algebra, and t a are Lie algebra elements. More convenient representation of the supersymmetry: � � d s a δ 2 |N ( λ a − s a λ | η a − s a χ ) e is a [ µ ˜ ¯ λ a ] j · t a , V a = V a = d σ a V a s a Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 11 / 38

Yang-Mills Amplitudes Yang-Mills Amplitudes Worldsheet correlation function N k MHV amplitudes as correlation function �� � V 1 . . . � A = V k V k + 1 . . . V n . Take exponential factors appearing in the vertex operators into the action to obtain the effective field equations � k ¯ ∂ σ Z = ¯ s i ( λ i , 0 , η i ) ¯ ∂ ( λ, µ, χ ) = δ ( σ − σ i ) , i = 1 � n � � � � ¯ ∂ σ W = ¯ ¯ µ, ˜ 0 , ˜ ˜ λ, ˜ = δ ( σ − σ p ) . ∂ χ s p λ p , 0 p = k + 1 ( Z , W ) are worldsheet spinors, thus unique solution � k � n � � 1 1 0 , ˜ Z ( σ ) = ( σ σ i ) ( λ i , 0 , η i ) , W ( σ ) = λ p , 0 . ( σ, σ p ) i = 1 p = k + 1 Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 12 / 38

Yang-Mills Amplitudes Yang-Mills Amplitudes �� � V 1 . . . � A = V k V k + 1 . . . V n Yang-Mills amplitudes � � n � k λ ( σ i )) � n a = 1 d 2 σ a i = 1 ¯ δ 2 (˜ λ i − ˜ p = k + 1 ¯ 1 δ 2 |N ( λ p − λ ( σ p )) . A = � n Vol GL ( 2 , C ) a = 1 ( a a + 1 ) where � k � n ˜ λ p λ i ˜ λ ( σ ) = ( σ, σ i ) , λ ( σ ) = ( σ, σ p ) . i = 1 p = k + 1 Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 13 / 38

Yang-Mills Amplitudes Yang-Mills Amplitudes �� � V 1 . . . � A = V k V k + 1 . . . V n Yang-Mills amplitudes � � n � k λ ( σ i )) � n a = 1 d 2 σ a i = 1 ¯ δ 2 (˜ λ i − ˜ p = k + 1 ¯ 1 δ 2 |N ( λ p − λ ( σ p )) . A = � n Vol GL ( 2 , C ) a = 1 ( a a + 1 ) where � k � n ˜ λ p λ i ˜ λ ( σ ) = ( σ, σ i ) , λ ( σ ) = ( σ, σ p ) . i = 1 p = k + 1 In particular, these tree-level scattering amplitudes localize fully on the support of the scattering equations contain only σ moduli, no additional moduli from the degree d of a line bundle are manifestly parity invariant. Yvonne Geyer (Oxford) Ambitwistor Strings for Four Dimensions September 23, 2014 13 / 38