Three-dimensional update Yu-tin Huang w CongKao Wen, Dan Xie, Henrik Johansson, Sangmin Lee, Henriette Elvang, Cynthia Keeler, Thomas Lam, Timothy M. Olson, Samuel Roland, David E Speyer National Taiwan University Oxford-Sept-2014 Yu-tin Huang National Taiwan University
Prelude Perturbation in a topological theory: Chern−Simons Matter Gravity+Matter AdA+AAA R Yu-tin Huang National Taiwan University
The suspect Chenr-Simons Matter: U(N) k × U(N) − k gauge fields ( A µ , ¯ A µ ) , 1. N = 6 (ABJM): SU(4) bi-fundamental matter ( φ I , ψ I , ¯ φ I , ¯ ψ I ) , I = 1 , 2 , 3 , 4 L = L C S + L φ, Kin + L ψ, Kin + L 4 φ 2 ψ 2 + L 6 φ 6 φ 4 + η I ψ I + 1 2 ǫ IJK η I η J φ K + 1 3 ! ǫ IJK η I η J η K ψ 4 , Φ( η ) = φ I + 1 ψ K + 1 ψ 4 + η I ¯ 2 ǫ IJK η I η J ¯ 3 ! ǫ IJK η I η J η K ¯ ¯ ¯ Ψ( η ) = φ 4 , A n (¯ 12 ¯ 3 · · · n )( λ, η ) A n ( 1 ¯ 23 · · · ¯ n )( λ, η ) Yu-tin Huang National Taiwan University
The suspect Chern-Simons Matter: U(N) k × U(N) − k gauge fields ( A µ , ¯ A µ ) , 1. N = 6 (ABJM): SU(4) bi-fundamental matter ( φ I , ψ I , ¯ φ I , ¯ ψ I ) , I = 1 , 2 , 3 , 4 L = L C S + L φ, Kin + L ψ, Kin + L 4 φ 2 ψ 2 + L 6 φ 6 SU(2) k × SU(2) − k gauge fields ( A µ , ¯ A µ ) , 2. N = 8 (BLG): SO(8) adjoint matter ( φ I v , ψ I c ) [ T a , T b , T c ] = f abc d T d Yu-tin Huang National Taiwan University
The suspect Chern-Simons Matter: U(N) k × U(N) − k gauge fields ( A µ , ¯ A µ ) , 1. N = 6 (ABJM): SU(4) bi-fundamental matter ( φ I , ψ I , ¯ φ I , ¯ ψ I ) , I = 1 , 2 , 3 , 4 SU(2) k × SU(2) − k gauge fields ( A µ , ¯ A µ ) , 2. N = 8 (BLG): SO(8) adjoint matter ( φ I v , ψ I c ) [ T a , T b , T c ] = f abc d T d Super Gravity: 1. N = 16 supergravity: Marcus, Schwarz 128 scalars are in the spinor representation of SO(16) ∈ E 8 , 8 /SO(16) → M n = 0 for odd n Yu-tin Huang National Taiwan University
The gauge theory: ABJM Yu-tin Huang National Taiwan University
Prelude The known for N = 4 SYM: The planar theory enjoys SU(2,2 | 4) DSCI The string sigma model enjoys fermionic self T-duality The (super)amplitude is dual to a (super)Wilson-loop The IR-divergence structure captured by BDS The leading singularities is given by residues of Gr ( k , n ) The amplitude has uniform trancsendentality The amplitudehedron Yu-tin Huang National Taiwan University
Prelude As comparison to ABJM: The planar theory enjoys SU(2,2 | 4) DSCI → OSp(6 | 4) The IR-divergence structure captured by BDS → Remarkably yes The leading singularities is given by residues of Gr ( k , n ) → OG(k,2k) The amplitude has uniform trancsendentality (*) → True so far Yu-tin Huang National Taiwan University
Known unknowns: 1. Why is the IR-divergence (Dual conformal anomaly equation) the same? Y-t, W. Chen, S. Caron-Huot � N � 2 A tree A 2 − loop 4 = BDS 4 4 k 2 � N � 2 � A tree A tree � � � �� ln u 2 6 , shifted A 2 − loop 6 = BDS 6 + R 6 + ln χ 1 + cyclic × 2 6 k 2 4 i u 3 At four-point to all orders in ǫ M. Bianchi, M. Leoni, S Penati , exponentiation verified at three-loops M. Bianchi, M. Leoni Yu-tin Huang National Taiwan University
Known unknowns: 2. Why is the amplitude non-analytic? → 3 3 5 3 4 6 2 a b 5 2 a b 2 a b 5 1 1 1 6 tree − i 6 2 3 3 5 + cyclic 1 1 → Yu-tin Huang National Taiwan University
Unknown knowns The string sigma model enjoys fermionic self T-duality → Unsucessful The (super)amplitude is dual to a (super)Wilson-loop → Unsucessful Yu-tin Huang National Taiwan University
Prelude Planar N = 4 SYM ∈ Gr ( k , n ) + Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, J. Trnka → → C α i ( f ) f 4 f 5 f 6 fc f 6 1 1 1 f 1 + 0 0 f 1 fa ( 1 + f 0 ) 1 + 1 / f 0 1 + 1 / f 0 f 2 f 1 f 2 f 6 fa C α i ( f ) = 1 1 0 1 f 3 + f 3 fb ( 1 + f 0 ) , 0 1 + 1 / f 0 1 + 1 / f 0 f 3 f 4 f 2 fb f 4 1 1 0 0 1 f 5 + 1 + 1 / f 0 1 + 1 / f 0 f 5 fc ( 1 + f 0 ) � � df i δ 4k | 4k ( C · W ) � A n = f i dia i Yu-tin Huang National Taiwan University
Prelude Planar N = 4 SYM ∈ Gr ( k , n ) + Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, J. Trnka → → C α i ( f ) f 4 f 5 f 6 fc f 6 1 1 1 f 1 + 0 0 f 1 fa ( 1 + f 0 ) 1 + 1 / f 0 1 + 1 / f 0 f 2 f 1 f 2 f 6 fa C α i ( f ) = 1 1 0 1 f 3 + f 3 fb ( 1 + f 0 ) , 0 1 + 1 / f 0 1 + 1 / f 0 f 3 f 4 f 2 fb f 4 1 1 0 0 1 f 5 + 1 + 1 / f 0 1 + 1 / f 0 f 5 fc ( 1 + f 0 ) � � df i δ 4k | 4k ( C · W ) � A n = f i dia i Yu-tin Huang National Taiwan University
Prelude Planar N = 4 SYM ∈ Gr ( k , n ) + Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, J. Trnka → → C α i ( f ) f 4 f 5 f 6 fc f 6 1 1 1 f 1 + 0 0 f 1 fa ( 1 + f 0 ) 1 + 1 / f 0 1 + 1 / f 0 f 2 f 1 f 2 f 6 fa C α i ( f ) = 1 1 0 1 f 3 + f 3 fb ( 1 + f 0 ) , 0 1 + 1 / f 0 1 + 1 / f 0 f 3 f 4 f 2 fb f 4 1 1 0 0 1 f 5 + 1 + 1 / f 0 1 + 1 / f 0 f 5 fc ( 1 + f 0 ) � � df i δ 4k | 4k ( C · W ) � A n = f i dia i Yu-tin Huang National Taiwan University
Conclusion The scattering amplitude of ABJM is given by integrals over cells in the positive orthogonal grassmannian OG k + Each cell in the positive orthogonal grassmannian OG k + → cell Gr ( k , 2k ) + . The canonical form has logarithmic singularity at ∂ OG k + Yu-tin Huang National Taiwan University
Orthogonal Grasmmannian Consider k -planes in n -dimensional space equipped with a symmetric bi-linear Q ij The orthogonal grassmannian ≡ Q ij C α i C β j = 0 Consider n = 2k and Q ij = η ij signature (+ , + , + , · · · , +) k = 1 , C α i = ( 1 , ± i ) � 1 � ± i cos z 0 − i sin z k = 2 , C α i = ± i sin z 0 1 i cos z � dC � A tree ( 1 · · · k ) · · · ( k · · · n − 1 ) δ ( Q ij C α i C β j ) δ 2k ( C · λ ) δ 3k ( C · η ) = n res S. Lee, D. Gang, E. Koh, E. Koh, A. Lipstein, Y-t Yu-tin Huang National Taiwan University
Orthogonal Grasmmannian Consider k -planes in n -dimensional space equipped with a symmetric bi-linear Q ij The orthogonal grassmannian ≡ Q ij C α i C β j = 0 Consider n = 2k and Q ij = η ij signature (+ , + , + , · · · , +) k = 1 , C α i = ( 1 , ± i ) � 1 � ± i cos z 0 − i sin z k = 2 , C α i = ± i sin z 0 1 i cos z � dC � A tree ( 1 · · · k ) · · · ( k · · · n − 1 ) δ ( Q ij C α i C β j ) δ 2k ( C · λ ) δ 3k ( C · η ) = n res S. Lee, D. Gang, E. Koh, E. Koh, A. Lipstein, Y-t Yu-tin Huang National Taiwan University
Positive Orthogonal Grasmmannian Positivity: ( i , i + 1 , · · · , i + k ) > 0 Q ij = η ij signature (+ , + , + , · · · , +) k = 1 , C α i = ( 1 , ± i ) � 1 ± i cos z − i sin z � 0 k = 2 , C α i = 0 ± i sin z 1 i cos z Yu-tin Huang National Taiwan University
Positive Orthogonal Grasmmannian Positivity: ( i , i + 1 , · · · , i + k ) > 0 Q ij = η ij signature (+ , + , + , · · · , +) k = 1 , C α i = ( 1 , ± i ) � 1 ± i cos z − i sin z � 0 k = 2 , C α i = 0 ± i sin z 1 i cos z Yu-tin Huang National Taiwan University
Positive Orthogonal Grasmmannian Positivity: ordered ( i , · · · , j ) > 0 Q ij = η ij signature (+ , − , + , · · · , − ) k = 1 , C α i = ( 1 , 1 ) � 1 � cos z 0 − sin z k = 2 , C α i = 0 sin z 1 cos z Positive for 0 ≤ z ≤ π/ 2 Volume form w. logarithmic singularity at the boundary: z = π/ 2, z = 0 dz cos z sin z = d log tan z � d log tan δ 4 ( C · λ ) δ 6 ( C · η ) This is not the amplitude A 4 ! Yu-tin Huang National Taiwan University
Positive Orthogonal Grasmmannian Positivity: ordered ( i , · · · , j ) > 0 Q ij = η ij signature (+ , − , + , · · · , − ) k = 1 , C α i = ( 1 , 1 ) � 1 � cos z 0 − sin z k = 2 , C α i = 0 sin z 1 cos z Positive for 0 ≤ z ≤ π/ 2 Volume form w. logarithmic singularity at the boundary: z = π/ 2, z = 0 dz cos z sin z = d log tan z � d log tan δ 4 ( C · λ ) δ 6 ( C · η ) This is not the amplitude A 4 ! Yu-tin Huang National Taiwan University
Positive Orthogonal Grasmmannian Positivity: ordered ( i , · · · , j ) > 0 Q ij = η ij signature (+ , − , + , · · · , − ) k = 1 , C α i = ( 1 , 1 ) � 1 � cos z 0 − sin z k = 2 , C α i = 0 sin z 1 cos z Positive for 0 ≤ z ≤ π/ 2 Volume form w. logarithmic singularity at the boundary: z = π/ 2, z = 0 dz cos z sin z = d log tan z � d log tan δ 4 ( C · λ ) δ 6 ( C · η ) This is not the amplitude A 4 ! Yu-tin Huang National Taiwan University
Branches of Positive Orthogonal Grasmmannian � 1 � cos z 0 − sin z k = 2 , C α i = 0 sin z 1 cos z � 1 � cos z 0 sin z k = 2 , C α i = − cos z 0 sin z 1 For 0 ≤ z ≤ π/ 2 Positivity: ( i , · · · , j ) > 0 and ± ( i , · · · , 2k ) > 0 � d log tan δ 4 ( C · λ ) δ 6 ( C · η ) + ( OG 2 + ) A 4 = The four-point amplitude is given by the sum of two branches in OG 2 + Yu-tin Huang National Taiwan University
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