A Penrose transform for the double copy Three routes to the double - PowerPoint PPT Presentation

A Penrose transform for the double copy Three routes to the double copy: Lie-polynomials, differential forms, and the worldsheet Lionel Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk QCD Meets gravity 13/12/2019 Continuing work with Hadleigh Frost, 1912.04198, building on ABHY: Arkani-Hamed, Bai, He, Yan, arxiv:1711.09102 , discussions w/ Francis Brown, Carlos Mafra, Ricardo Monteiro, Oliver Schlotterer & an after dinner talk by Kapranov 2012. cf also related work by Sebastian Mizera 1912.03397 and Song He.

Introduction This talk develops three interlocking mathematical structures that underpin the double copy. 1 Lie polynomials and their homomorphisms to numerators. 2 The ABHY [Arkani-Hamed, Bai, He, Yan] geometry of ( n − 3 ) -forms on K n = R n ( n − 3 ) / 2 , the space of Mandelstams s ij . 3 The geometry of M 0 , n the moduli space of n -marked points on CP 1 . We further develop a twistorial correspondence between 2 & 3: ( s ij , σ i ) ∈ Y n p ւ ց q T ∗ s ij ∈ K n D M 0 , n ∋ ( τ i , σ i ) giving a transform from CHY/Ambitwistor-string half-integrands to scattering forms etc..

Words and Lie polynomials Free Lie algebras, [Reutenauer 1993] For n particles: let W ( n − 1 ) = R ( n − 1 )! with basis n − 1-words in n − 1 distinct letters x 1 x 2 . . . x n − 1 ↔ i.e. permutations S n − 1 . . Definition The Lie polynomials Lie ( n − 1 ) ⊂ W ( n − 1 ) are spanned by • Lie monomials Γ ∈ Lie ( n − 1 ) , combinations of n − 1 -words made of complete commutators Γ = [[[ x 1 , x 3 ] , [ x 4 , x 5 ]] , x 2 ] so [ a , b ] = ab − ba is skew and satisfies Jacobi. • An oriented connected trivalent tree graph Γ with n − 1 4 5 3 2 leaves rooted at n, i.e. n = 6 : 1 6

Duality for Lie polynomials Theorem (Radford 1979) Lie ( n − 1 ) = R ( n − 2 )! with ‘DDM basis’ half-ladders/combs: a 3 a n − 1 a 2 Γ 1 a := n 1 Where a = a 2 a 3 . . . a n − 1 is a permutation of 2 , . . . , n − 1 . • W ( n − 1 ) = R ( n − 1 )! has inner product ( a , b ) with distinct words giving an orthonormal basis. • Lie ( n − 1 ) ⊂ W ( n − 1 ) by expanding all [ x i , x j ] = x i x j − x j x i . • For word a , (Γ , a ) = coefficient of a in expansion of Γ . • (Γ , a ) = ± 1 iff Γ planar for ordering a with ± orientation. Theorem (Ree 1958) c ∈ Lie ( n − 1 ) ⊂ W ( n − 1 ) ⇔ ( c , a ✁ b ) = 0 ∀ nontrivial a , b (now of different sizes | a | + | b | = n − 1 )

Geometry of Mandelstam space K n • Given n null momenta k i , � k i = 0, set s ij = ( k i + k j ) 2 n � s ij = 0 } = R n ( n − 3 ) / 2 . K n = { s ij = s ji | s ii = 0 , and j = 1 • Factorization hyperplanes: given by s I = 0 where � 2 �� � s I := s ij = k i , I ⊂ { 1 , 2 , . . . , n } . i , j ∈ I i • s I = 0 and s I ′ = 0 compatible iff I ′ ⊂ I or I c , complement. • Maximal compatible sets are in 1:1 correspondence with the n − 3 propagators of trivalent diagrams Γ .

Abstract biadjoint scalar The p th propagator of graph Γ carries momentum � i ∈ I p k i , I p ⊂ { 1 , . . . , n } giving propagators: 1 1 � = , s I = s ij � n − 3 d Γ p = 1 s I p i < j ∈ I Definition Abstract biadjoint scalar has amplitudes Γ ⊗ Γ (Γ , a )Γ � ∈ ⊗ 2 Lie ( n − 1 ) , � m = m a = ∈ Lie ( n − 1 ) d Γ d Γ Γ Γ Standard biadjoint scalar is m ( a , b ) = ( m a , b ) . Dress abstract m with numerators to obtain favourite theories.

BCJ numerators BCJ numerators { N Γ } give homorphism N : Lie ( n − 1 ) → V : Γ s + Γ t + Γ u = 0 ⇒ N Γ s + N Γ t + N Γ u = 0 2 3 2 3 3 2 − + = 0 1 4 1 4 1 4 and for embeddings into larger diagrams. Examples: • Colour ordering: Γ → (Γ , a ) ∈ W ( n − 1 ) . • For g i ∈ g , a Lie algebra: Γ → c Γ = tr ( g n Γ( g 1 , . . . , g n − 1 )) ∈ { invt polys on g n } . • Kinematic numerators: [Du, Feng Tei,. . . ] Γ → N k ,ǫ ∈ { invt polys in ( k i , ǫ i ) } . Γ Do all such linear maps arise from choice of Lie algebra?

Theories with different combinations of numerators Fu, Du, Huang, Feng Tei, 2017, BCJ, Chiarodoli, Roiban, 1909.01358, cf also CHY, CGMMRS 1506.08771 N l Γ N r � Γ Amplitudes: M = d Γ Γ N r N k ,ǫ N k , k N k ,ǫ, m N k ,ǫ, g c Γ or (Γ , a ) Γ Γ Γ Γ N l N k ,ǫ E Γ N k , k BI Galileon Γ N k ,ǫ, m EM DBI EMS Γ U ( 1 ) m U ( 1 ) m × U ( 1 ) ˜ m N k ,ǫ, g EYM ext. DBI EYMS EYMS Γ SU ( N ) × U ( 1 ) ˜ SU ( N ) × SU (˜ m N ) c Γ or (Γ , a ) YM Nonlinear σ EYMS gen . YMS Biadjoint Scalar SU ( N ) × U ( 1 ) ˜ m SU ( N ) × SU (˜ SU ( N ) × SU (˜ N ) N ) Table: Theories arising from the different choices of numerators.

( n − 3 ) -forms on K n —the ABHY scattering forms ABHY construct homomorphism Lie ( n − 1 ) ≃ Ω n − 3 K n ⊂ Ω n − 3 K n s • Given Γ define w Γ = ( − 1 ) Γ � n − 3 p = 1 ds I p • They prove w Γ s + w Γ t + w Γ u = 0 so w Γ provide numerators w Γ : Lie ( n − 1 ) → Ω n − 3 K n ⊂ Ω n − 3 K n . s • Given other numerators N Γ , define scattering forms N Γ w Γ � ∈ Ω n − 3 Ω N = K n . s d Γ Γ E.g. Ω a = Ω (Γ , a ) when N Γ = (Γ , a ) . • Ω N is projective on K n ⇔ s , t , u relations on N Γ . ∂ � N Γ s + N Γ t + N Γ u = 0 ⇔ Υ � Ω N = 0 , Υ = s ij . ∂ s ij ij

M 0 , n and its boundary divisor Define: Deligne-Mumford compactification σ 2 σ 1 M 0 , n = {× n CP 1 − diagonals } / Möbius σ n with coordinates • ( σ 1 , . . . , σ n ) , fix Möbius by ( σ 1 , σ n − 1 , σ n ) = ( 0 , 1 , ∞ ) . u i j = σ i j − 1 σ j i − 1 • Or planar cross-ratios: σ i j σ i − 1 j − 1 , σ i j = σ i − σ j . ¯ I I Boundary ∂ M 0 , n = D = ∪ I D I labelled by I ⊂ { 1 , . . . , n } . 1 • Planar Γ has propagators s Ip ↔ I p = { i p , . . . , j p − 1 } , • Gives n − 3 cross ratio u I p := u i p j p − 1 coords for M 0 , n s.t. D I p = { u I p = 0 }

Lie polynomials in M 0 , n Proposition H n − 3 ( M 0 , n − D ) = Lie ( n − 1 ) generated by Γ → T Γ = {| u I p | = ǫ, p = 1 , . . . n − 3 } . tori around the n − 3 -fold intersection points ↔ Γ of the D I . Proof: M 0 , 4 = CP 1 and D = { Γ s , Γ t , Γ u } T Γ t T Γ s T Γ u 2 3 2 3 3 2 − + 1 = 0 1 4 1 4 4 ↔ . This picture embeds in M 0 , n ∀ n , (Γ s , Γ t , Γ u ) . Corollary Γ(Ω n − 3 M 0 , n ) = H n − 3 ( M 0 , n − D ) = Lie ( n − 1 ) ∗ . D � i d log σ ˜ ai ˜ ai − 1 ˜ generated by Parke-Taylors: PT a = , a = an . vol ( SL ( 2 ))

The correspondence between K n and T ∗ D M 0 , n Lemma K n = { Space of sections of T ∗ D M 0 , n } Proof: Let τ i be fiber coords on T ∗ D M 0 , n , so τ = � i τ i d σ i . Sections are s ij � � � τ = s ij d log σ ij = E i d σ i , E i := . ✷ σ ij ij i j • E i = 0 are the scattering equations. • Incidence equations are τ i = E i ( s kl , σ j ) . • These are incidence equations of a twistor correspondence K n × M 0 , n = Y n ∋ ( s ij , σ i ) p ւ ց q T ∗ s ij ∈ K n D M 0 , n ∋ ( τ i = E i ( s kl , σ j ) , σ i )

CHY formulae as a Penrose transform K n × M 0 , n = Y n ∋ ( s ij , σ i ) p ւ ց q T ∗ s ij ∈ K n D M 0 , n ∋ ( τ i = E i ( s kl , σ j ) , σ i ) • The Penrose transform by p ∗ q ∗ i.e.: • The CHY formulae are � q ∗ � δ ( τ ) n − 3 � I l I r ¯ M ( s ij , . . . ) = M 0 , n = p − 1 ( s ij ) • Here I l , I r ∈ Ω n − 3 ( M 0 , n ) are CHY half-integrands but also often depending also on polarization data etc., • E.g., LHS = m ( a , b ) for ( I l , I r ) = ( PT a , PT b ) . • There is an empirical direct correspondence between choices of I l / r and numerators N Γ .

The forms w Γ from the symplectic volume form • Let ω = d τ i ∧ d σ i , then ω n − 3 has top degree and q ∗ ω n − 3 ∈ Ω n − 3 K n ⊗ Ω n − 3 M 0 , n . D • Define/evaluate the ABHY forms by n − 3 � � w Γ := ( − 1 ) Γ ω n − 3 . ds I p = T Γ p = 1 • Thus w Γ s + w Γ t + w Γ u = 0 follows from T Γ s + T Γ t + T Γ u = 0. • We can expand ω n − 3 as ω n − 3 = � w Γ 1 a ⊗ PT 1 a . a ∈ S n − 2

CHY half-integrands, scattering forms and numerators • Just as w Γ give numerators to give scattering forms, ω n − 3 gives CHY half-integrand for scattering forms � q ∗ I l ω n − 3 ¯ δ n − 3 ( τ i ) ∈ Ω n − 3 Ω I l = ( K n ) s M 0 , n • Projectivity follows from ∂ ∂ � ω n − 3 = � ω n − 3 = � � � Υ = τ i τ i d σ i s ij ∂ s ij ∂τ i ij i i which vanishes against the delta functions. • Thus N I l Γ w Γ � Ω I l = d Γ Γ with N I l Γ satisfying stu -relations by ABHY.

An invariant definition of associahedral n − 3-planes. Correspondence: planar Γ ↔ P Γ associahedral n − 3-plane: • Choose ordering and planar factorization channels I ij = { i , i + 1 , . . . , j − 1 } associated x I ij = � i ≤ l < m < j s lm . • n − 3 ∂ ∂ � � P Γ = D I p , D I := − . ∂ X I ∂ X J J ∈ I c p = 1 I c = { planar factorization channels incompatible with I } , I c ij = { lines that cross the line from i to j } . • The ABHY P a = P Γ 1 a . P a � ω n − 3 = PT a P Γ � w Γ ′ = (Γ ′ , a ) . so Ex: biadjoint scalar follows via CHY as � � q ∗ PT a P b � ω n − 3 ¯ δ n − 3 ( τ i ) = q ∗ PT a PT b ¯ δ n − 3 ( τ i ) P b � Ω PT a = M 0 , n M 0 , n = m ( a , b ) .