ABJM Amplitudes and! Orthogonal Grassmannian

Transcription

1 ABJM Amplitudes and! Orthogonal Grassmannian Sangmin Lee Seoul National University 12 June 2014 Recent Developments in Theoretical Physics, KIAS

2 Introduction

3 Scattering Amplitudes in Gauge Theories... beyond Feynman diagrams

4 Scattering amplitudes in Twistor string theory MHV vertices expansion BCFW recursion relation Amplitude/Wilson-loop duality BCJ color/kinematics duality [Witten 03] [Cachazo,Svrcek,Witten 04] [Britto,Cachazo,Feng 04] [Britto,Cachazo,Feng,Witten 05] [Drummond,Henn,Korchemsky,Sokatchev 06-08] [Alday,Maldacena 07][Berkovits,Maldacena 08] [Beisert,Ricci,Tseytlin,Wolf 08] [Bern,Carrasco,Johansson 08] Polylogarithms, symbols, Hopf algebras [Goncharov,Spradlin,Vergu,Volovich 10] [Duhr 12] Grassmannians, on-shell diagrams [Arkani-Hamed,Cachazo,Cheung,Kaplan 09] [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12] Amplituhedron [Arkani-Hamed,Trnka 13]

5 Old ideas, reincarnated 1960s : S-matrix program 1980s Spinor-helicity formulation : p aȧ = l a lȧ Grassmannian Recursion relations (Parke-Taylor, Berends-Giele, ) Current algebra / twistor (Nair) KLT relation : closed = (open) 2 BCFW Twistor String BCJ a 0! 0 limit of string theory Multiple zeta

6

7 Scattering Amplitudes of ABJM theory D = 4 N = 4 SYM D = 3 N = 6 SCS AdS5 x S5 AdS4 x CP3

8 This talk is based on S.L. [ ] Dongmin Gang, Yu-tin Huang, Eunkyung Koh, S.L., Arthur E. Lipstein [ ] Yu-tin Huang, S.L. [ ] Yu-tin Huang, Henrik Johansson, S.L. [ ]! Joonho Kim, S.L. [ ]

9 Kinematics

10 Momentum conservation linearized - 4d Momentum conserving delta-function d (4) (P), P = n  i=1 p i Spinor-helicity formulation p aȧ = p µ (s µ ) aȧ = lȧl a =) P aȧ = n  i=1 l iȧ l i a BCFW deformation leads to a recursion formula l i! S i jl j, l i! l j (S 1 ) j i, S 2 GL(n, C) Grassmannian integral is based on [Britto, Cachazo, Feng, Witten 04-05] C mi l i = 0, 9 a m, l i = a m C mi, C mi 2 G(k, n) [Arkani-Hamed,Cachazo,Cheung,Kaplan 09]

11 Momentum conservation linearized - 3d Momentum conserving delta-function d (3) (P), P = 2k  i=1 p i Spinor-helicity formulation p ab = p µ (s µ ) ab = l a l b =) P ab = 2k  i=1 l i al i b BCFW deformation leads to a recursion formula l i! R i jl j, R 2 O(n, C) [Gang,Huang,Koh,SL,Lipstein 10] Grassmannian integral is based on C mi l i = 0 & 9 a m, l i = a m C mi =)  i C mi C ni = 0, C mi 2 OG(k,2k) [SL 10]

12 On-shell super-fields SYM4 SU(N) gauge SU(4) R F = A + + h I c I hi h J f IJ e IJKLh I h J h K c L e IJKLh I h J h K h L A helicity SU(4) R SCS6 [U(N) U(N)] gauge SU(4) R Z : (N, N;4), : (N, N; 4). F = f 4 + h I y I e IJKh I h J f K e IJKh I h J h K y 4, F = ȳ 4 + h I f I e IJKh I h J ȳ K e IJKh I h J h K f 4. ( )=(4+4) h I, h J = d J I SO(6) Clifford algebra, only U(3) manifest.

13 Super-conformal symmetry Lorentz: d = 4, N = 4 SYM SO(1, 3) d = 3, N = 6 SCS SO(1, 2) Conformal: SO(2, 4) ' SU(2, 2) SO(2, 3) ' Sp(4, R) Super- Conformal: PSU(2, 2 4) SU(4) R OSp(4 6) SO(6) R Super-charges: 32 24

14 Grassmannian Integral for ABJM amplitudes [SL 1007] [Arkani-Hamed,Cachazo,Cheung,Kaplan 0907]

15 Grassmannian Integral formula in 3d [SL 1007] Z A tree 2k (L) = d k 2k C vol [GL(k)] d k(k+1)/2 (C C T ) d 2k 3k (C L) M 1 M 2 M k 1 M k 2k C = k GL(k) O(2k) (i) (i + k 1) (C L) m = C mi L i 2k: total number of external legs (C C T ) mn = C mi C ni M i = e m 1 m k C m1 (i) C m 2 (i+1) C mk (i+k 1)

16 Grassmannian Integral formula in 3d Z A tree 2k (L) = d k 2k C vol [GL(k)] d k(k+1)/2 (C C T ) d 2k 3k (C L) M 1 M 2 M k 1 M k Superconformal symmetry L L! manifest 2 L L! killed by d(c CT ) C T bc + bc T C = I 2k 2k LL! L T (C T bc + bc T C) L = 0 Cyclic symmetry C C T = 0 =) M i M i+1 =( 1) k 1 M i+k M i+1+k

17 6-point amplitude [Bargheer-Loebbert-Meneghelli 1003] (6-scalar) color-stripping reduces number of diagrams (6-fermion)

18 6-point amplitude [Bargheer-Loebbert-Meneghelli 1003] 6-scalar 6-fermion

19 6-point amplitude from Grassmannian integral Factorization gauge C = c 1 14 c 15 c c 24 c 25 c 26 A c 34 c 35 c 36 6-fermion amplitude f A 6y = +i i (h13ih46i + ih2 p 123 5i) 3 (p 123 ) 2 (h23ih56i + ih1 p 123 4i)(h12ih45i + ih3 p 123 6i) (h13ih46i ih2 p 123 5i) 3 (p 123 ) 2 (h23ih56i ih1 p 123 4i)(h12ih45i ih3 p 123 6i). shows the factorization channel clearly: Check via recursion relation! A 6 (1, 2, 3, 4, 5, 6)! A 4 (1, 2, 3, f ) 1 (p 123 ) 2 A 4( f, 4, 5, 6)+(regular).

20 On-shell diagrams and Positive Grassmannian [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12] [Huang, Wen 1309][Kim, SL, 1402][Huang, Wen, Xie 1402]

21 On-shell diagrams [Arkani-Hamed,Bourjaily,Cachazo,Gonchaorv,Postnikov,Trnka 12] ~ G(1,3) ~ G(2,3)

22 On-shell diagrams and permutations [Arkani-Hamed,Bourjaily, Cachazo,Gonchaorv,Postnikov,Trnka 12] Build up big Grassmannian from tiny Grassmannians G(1,3) and G(2,3) through BCFW bridging! =

23 On-shell diagrams in 3d [Joonho Kim, SL 1402][Huang, Wen1309] Building blocks: d 3 (P)d 6 (Q) h14ih34i 4-point on-shell amplitude Z d 2 ld 3 h unique integral preserving superconformal symmetry

24 On-shell diagrams in 3d [Joonho Kim, SL 1402][Huang, Wen1309] On-shell diagrams = pairing of external particles Yang-Baxter -like equivalence relation ~ non-uniqueness of Euler-angles for an SO(3) matrix.

25 BCFW bridging 2 O(2, C)

26 BCFW = OG(2,4)

27 Geometry of OG(k,2k) Complex OG(k) = moduli space of null k-planes in C 2k. = O(2k)/U(k) Ex) k = 2 : k = 3 : O(4)/U(2) CP 1 O(6)/U(3) CP 3

28 Real slices of OG(k) We cannot use component-wise real C : C C T = 0 Reality conditions should be imposed on (gauge invariant) Plücker coordinates. Several inequivalent real slices can be defined. Ex) k = 2 :

29 Positive OG(k) Positive OG is a subset of a real slice of OG Canonical choice: [Huang, Wen1309] Wick-rotate to alternating signature, (, +,, +,,, +) C h C T = 0 and demand that all ordered minors be positive!

30 Classification of on-shell diagrams Barring bubbles, the number of crossing ( level ) is well-defined. Top diagram (maximal crossing) is unique modulo Yang-Baxter equivalence. Level of top cell = k(k 1) 2 Level of diagram contributing to tree amplitude = (2k 3)

31 Classification of on-shell diagrams Bottom diagrams! (no crossing) Number of bottom diagrams is the k th Catalan number:

32 OG tableaux - I. Unfolded [Joonho Kim, SL 1402] Mapping to a upper-right-triangle tableaux with Yang-Baxter ambiguity fixed. Cyclic symmetry spontaneously broken. k(2k 1) boxes.

33 OG tableaux - I. Folded Projection to a sub-square tableaux with k 2 or less boxes. Invertible - no information is lost through projection

34 OG tableaux for k = 2 Diagrams sharing the same empty tableaux are grouped together. Level (number of crossing) We will call each tableau a cell with the OG geometry in mind. Source column

35 OG tableaux for k = 3

36 Canonically positive coordinates Mapping from tableaux to OG matrices. Use GL(k) gauge symmetry to fix the matrices in row-echelon form The source columns, when separated, form a (k x k) identity matrix.

37 Bottom cells Given a folded tableau,! there is a unique source-sink connection with no crossings. The absolute value of sink element is fixed by orthogonality. The sign is fixed uniquely by positivity: Ex) k = 2

38 Canonical BCFW bridging Source columns remain fixed. Sink columns are bridged by (2x2) matrices from the right by Ex) k = 3, level 2 : Sign fixed by positivity (t > 0).

39 Canonical BCFW bridging Ex) k = 3, level 3 (top) :

40 Integration measure Contour problem for tree amplitude is solved A 8 =

41 Combinatorics of OG tableaux T k (q) = k(k 1)/2 Â l=0 T k,l q l Closed-form formula known (!)

42 Polytope interpreted as the Euler characteristic of the OG. Conjecture: the polytope has the topology of a ball for all k.

43 Summary

44 Summary BCFW-like recursion relation Grassmannian integral formula / on-shell diagram Loops, strings Twistor-string-like integral formula, KK/BCJ relations Dual conformal symmetry (Yangian) Momentum twistor, amplituhedron? Amplitude/Wilson-loop/Correlator triality?

45 Thank you