1. PRIN meeting - Pisa, 17/5/2013 S. Penati 1 A Survey in ABJM: Scattering Amplitudes and Wilson loops Silvia Penati University of Milano-Bicocca and INFN A Survey in ABJM: Scattering Amplitudes and Wilson loops Silvia Penati University of Milano-Bicocca and INFN In collaboration with Marco Bianchi, Gaston Giribet, Matias Leoni, Andrea Mauri, Alberto Santambrogio

2. The N=6 SCFT (ABJM) PRIN meeting - Pisa, 17/5/2013 S. Penati 2 SU(2) x SU(2) global symmetry AB The theory is at its superconformal point Aharony, Bergman, Jafferis, Maldacena 0806.1218 Aharony, Bergman, Jafferis 0807.4924

3. AdS4/CFT3 CORRESPONDENCE PRIN meeting - Pisa, 17/5/2013 S. Penati 3 For N << k type IIA on AdS x CP 5 43 N >> k (Bagger, Lambert; Gustavsson; Aharony, Bergman, Jafferis, Maldacena)

4. Why AdS4/CFT3 correspondence? The ultimate goal is to go deeper inside the nature of the AdS/CFT correspondence by disclosing model independent features. (Not simple replication of the 4d case) Why scattering amplitudes ? The main focus is on conceivable integrability structures at both sides of the correspondence. Tracks in the on-shell sector of the theories Why Wilson loops? Light-like Wilson loops are strictly connected to the amplitudes, via AdS/CFT correspondence. BPS WL computed using localization. PRIN meeting - Pisa, 17/5/2013 S. Penati 4

5. Plan of the talk 1) Amplitudes in N=6 Chern-Simons-matter models (ABJM/ABJ models): Duality with light-like 4pt polygon WL, Dual superconformal invariance, BDS-like ansatz for 4pt 2) ½ BPS Wilson loops for ABJM and comparison with results from localization PRIN meeting - Pisa, 17/5/2013 S. Penati 5

6. PRIN meeting - Pisa, 17/5/2013 S. Penati 6 Dualities Symmetries IntegrabilityIntegrability of the planar on-shell sector of the theory AdS/CFT perturbative investigation In N=4 SYM the planar on-shell sector exhibits powerful symmetries and dualities that rely on an integrable structure at both sides of AdS/CFT correspondence

7. DUALITIES DUALITIES PRIN meeting - Pisa, 17/5/2013 S. Penati 7 CFT contains other non-trivial objects like Wilson loops and correlation functions - Amplitudes are functions of p, p = 0 - Polygon WL and correlators are functions of x ii 2 Dual variables ij

8. What happens in three dimensions ? String side: Integrability is still an open question, although classically there are strong evidences. CFT side: All loop Bethe Ansatz for the dilatation operator. But effects on the on-shell sector still to be deeply investigated at both sides of the correspondence. PRIN meeting - Pisa, 17/5/2013 S. Penati 8

9. 1) Planar scattering amplitudes Only matter particles are dynamical. Therefore only scattering of scalars and fermions in the bifundamental repr. Gauge invariance, Lorentz and SU(2)xSU(2) invariances force the number of particles to be even and configurations with equal number of bifundamental and anti-bifundamental fields PRIN meeting - Pisa, 17/5/2013 S. Penati 9 N MHV n/2-2

10. Tree level results Tree level results General constraints from superconformal invariance, once solved, allow to determine tree level amplitudes in terms of a restrict number of functions ( Bargheer, Loebbert, Meneghelli ). The dual superconformal symmetry has been proved at tree level ( Huang, Lipstein ). The Yangian symmetry is realized at tree level. Proposed generating functional for tree amplitudes is manifestly Yangian invariant ( Lee ). PRIN meeting - Pisa, 17/5/2013 S. Penati 10

11. We move at loop level… Two loops We move at loop level… Two loops PRIN meeting - Pisa, 17/5/2013 S. Penati 11 JHEP 1201 (2012) 056, JHEP 1112 (2011) 073

12. (1) Amplitude/WL duality Duality with a light-like 4-polygon Wilson loop It is zero at one-loop, but…. PRIN meeting - Pisa, 17/5/2013 S. Penati 12 (Henn, Plefka, Wiegandt; Bianchi, Leoni, Mauri, SP, Santambrogio; Bianchi, Giribet, Leoni, SP, 1304.6085) x x x x 1 2 3 4

13. (2) Dual conformal invariance PRIN meeting - Pisa, 17/5/2013 S. Penati 13 Chen-Huang (JHEP 11(2011) 057) have computed the 2 loop correction by assuming dual conformal invariance + generalized unitarity cuts They find exactly the same result !!

14. (3) Another duality ?? Identification: PRIN meeting - Pisa, 17/5/2013 S. Penati 14 At all orders in epsilon : Bianchi, Leoni, SP, arXiv:1112.3649

15. (4) An all loop 4pt amplitude N=4 SYM case the BDS ansatz Analogy between the integrable structures ( Gromov, Vieira ) Identification of 4D 4pt at 1 loop and 3D 4pt at 2 loops PRIN meeting - Pisa, 17/5/2013 S. Penati 15 Therefore…..

16. PRIN meeting - Pisa, 17/5/2013 S. Penati 16 Confirmed by our two-loop result Confirmed at strong coupling 4pt amplitude at four loops needs be computed !!

17. PRIN meeting - Pisa, 17/5/2013 S. Penati 17 2) 1/2 BPS vs 1/6 Wilson loops Superconnection We consider U(N) x U(M) ABJ(M) theory (Drukker-Trancanelli 0912.3006) C = circle

18. PRIN meeting - Pisa, 17/5/2013 S. Penati 18 Prediction from localization Pestun 0712.2824; Kapustin-Willett-Yaakov 0909.4559; Drukker-Marino-Putrov 1007.3837 Drukker-Trancanelli

19. PRIN meeting - Pisa, 17/5/2013 S. Penati 19 Change of framing Guadagnini-Martellini-Mintchev NPB330 (1990) 575 Framing gives an overall phase 1loop

20. PRIN meeting - Pisa, 17/5/2013 S. Penati 20 The two-loop calculation (DRED scheme) Bianchi, Giribet, Leoni, SP, 1303.6939; in preparation

21. Conclusions and outlook Conclusions and outlook Amplitudes: already remarkable results but still…..many open problems 4pt amplitude at four loops to confirm the BDS-like ansatz and amplitude/WL duality. Are scattering amplitudes with n>4 dual to some Wilson loop? Which one? Better comprehension of the holomorphic anomaly arising for n>4. Correlators? We have generalized the calculation to the case of ABJ theory. At the order we have worked things do not seem to change much. What about higher orders? Fermionic T-duality not completely understood at strong coupling Wilson Loops: We have found perfect agreement with the two-loop result from localization The calculation has required a consistent effort in evaluating loop integrals We could exploit the techniques developed for evaluating more general BPS Wilson loops ( Cardinali, Griguolo, Martelloni, Seminara, 1209.4032 ) PRIN meeting - Pisa, 17/5/2013 S. Penati 21

22. PRIN meeting - Pisa, 17/5/2013 S. Penati 22 New COST Action “The String Theory Universe” 17 european Countries, 130 people and more… http://www.cost.eu/domains_actions/mpns/Actions /MP1210