1. Anastasia Volovich Brown University Miami, December 2014 Golden, Goncharov, Paulos, Spradlin, Vergu

2. Scattering Amplitudes The bread and butter of quantum fields theories, such as QCD, both theoretically and experimentally are scattering amplitudes. The last few years have seen a lot of progress in our understanding of the structure of scattering amplitudes and in our ability to do computations both for theoretical and phenomenological purposes. Remarkable results range from precision predictions in QCD which are important for understanding the LHC data to the discovery of new symmetries in gauge and gravity theories.

3. Scattering amplitudes play important role in many interrelated subjects. They bring together the community of Amplitudeologists. Amplitudes 2009, Durham; Amplitudes 2010, London; Amplitudes 2011, Ann Arbor; Amplitudes 2012 Hamburg; Amplitudes 2013 Tegernsee; Amplitudes 2014 Paris; Amplitudes 2015 Zurich

4. The goals of this program to explore the rich mathematical structures of scattering amplitudes to exploit these structures as much as possible to compute amplitudes

5. Many have contributed to recent developments including: Alday, Arkani-Hamed, Basso, Badger, Bargheer, Beisert, Belitski, Bern, Berkovits, Boels, Bourjaily, Brandhuber, Brodel, Bjerrum-Bohr, Bullimore, Britto, Cachazo, Caron-Huot, Carrasco, Cheung, Damgaard, Dennen, Del Duca, Duhr, Dixon, Dolan, Duhr, Drummond, Eden, Ellis, Elvang, Fend, Ferarra, Ferro, Forde, Forini, Franko, Freedman, Gangl, Gaiotto, Goddard, Goncharov, Green, He, Henn, Heslop, Hodges, Huang, Huber, Ita, Johnannson, Kallosh, Kaplan, Khoze, Kiermaier, Kraimer, Kristjansen, Korchemsky, Kosower, Kuntz, Lipatov, Loebbert, Melnikov, Maitre, Mafra, Maldacena, Mason, Monteiro, Naculich, O’Connell, Nastase, Papathanasiou, Plefka, Prygarin, Paulos, Roiban, Rosso, Sabio Vera, Schnizer, Schloterer, Schwab, Sever, Skinner, Smirnov, Sokatchev, Spence, Spradlin, Staudacher, Stelle, Stieberger, Svrcek, Taylor, Travaglini, Trnka, Tye, Vanhove, Vergu, Vieira, Yang, Wen, Weinzel, Witten, Zanderighi

6. Goal of My Talk explore cluster algebra structure of scattering amplitudes in planar N=4 Yang-Mills explain how to use it to compute 2 and 3 loop amplitudes

7. Planar 2-loop 6-point MHV N=4 SYM Arkani-Hamed: N=4 YM is 21 st century harmonic oscillator [Bern, Dixon, Kosower, Roiban, Spradlin, Vergu, AV] Goncharov, Spradlin, Vergu, AV Is there a structure? How to generalize this formula? Why do these particular arguments appear? Why only classical polylog functions? Cluster algebraic structure is the key

8. Cluster Algebras Cluster algebras were first discovered and developed by Fomin and Zelevinski (2002). Very informally: commutative algebras constructed from distinguished generators (cluster variables) grouped into disjoint sets of constant cardinality (clusters) which are constructed recursively from the initial cluster by mutations. Cluster algebra portal: http://www.math.lsa.umich.edu/~fomin/cluster.html

9. Cluster Algebra Consider a collection of variables subject to the exchange relation Seed exchange relation with initial cluster -- cluster coordinates forming cluster algebra -- mutation represent using quivers

10. Quivers and Mutations We can define cluster algebra by a quiver: oriented graph without loops and 2-cycles. Given a quiver, get a new one by mutation rule: For vertex 1:

11. Quivers and Cluster Coordinates We can encode a quiver by a skew-symmetric matrix To each vertex i associate variable Use matrix b to define mutation relation at vertex k In practice to construct all cluster variables from a given quiver see Keller Java program

12. --Amplitudes are functions on --Scott (2003) classified all Grassmannian cluster algebras of finite type. 3 x (n-5) initial quiver. Grassmanian cluster algebras

13. cluster algebra Start with quiver. Generate all coordinates by mutations. Mutation generates 14 clusters. 15 A-coordinates: 6 fixed ; 9 unfixed 15 X-coordinates: Note: The top 9/15 are exactly the arguments in 2-loop 6-point amplitude with Golden, Goncharov, Spradlin, Vergu

14. cluster algebra 49 A-coordinates: 35 Plucker brackets + cyclic =14 Mutations generate 833 clusters Stasheff polytope: 833 V, 2499 E, 2856 F2 (1785S+ 1071P), 1547 F3 Analyzing all quivers: 385 X-coordinates

15. by D. Parker

16. Cluster Structure in Amplitudes Symbols: all n-point amplitudes in SYM theory have symbol alphabet with subset of cluster A-coordinates on Gr(4,n)  Cluster Bootstrap in Mark’s talk Coproduct: -- for two-loop MHV amplitudes, only cluster X- coordinates appear (with particular Poisson brackets) -- for higher-loop or non-MHV, not yet understood Functions: there is a particular class of natural ``cluster polylogarithm’’ functions which exhibit these properties

17. 2. Coproduct: 2-loop 7-point MHV

18. All and in coproduct for 2-loop 7-points amplitude are cluster X-coordinates for cluster algebra Out of 385 only 231 appear in the amplitude. What is the criterion?? [Note: 9/15=231/385!] For each are in the same cluster. Appear in pairs with zero Poisson bracket. is a sum of 42 squares of Stasheff polytope. with Golden, Goncharov, Spradlin, Vergu

19. 3. Function: Cluster Polylogarithm In order to find the corresponding function, we need to find a function whose coproduct can be expressed entirely in terms of cluster coordinates

20. 3. Function: Cluster Polylogarithm To find functions, all one has to do is solve There is a unique solution for cluster algebra

21. 3. Function: Cluster Polylogarithm Recall that derived from the amplitude side only has pairs with Poisson bracket zero, which is an additional constraint, and leads to a particular combination of pentagon functions.

22. 2. Function: other Cluster Polylogarithms All non-trivial degree 4 cluster functions for are linear combinations of functions With D. Parker and A. Scherlis

23. 2. Function: 2-loop 7-point amplitude Cluster polylog functions are building blocks necessary to write down all-n function for 2-loop MHV. with Golden, Spradlin, Paulos

24. Conclusion We have advocated the study of cluster structure of N=4 YM amplitudes. We can use this structure for advancing computations: 2 and 3 loop MHVs. Many questions remain: cluster structure @ higher loops, other helicities, strong coupling….. very impressive explicit results by Dixon, Drummond, Duhr, Henn, Pennington, von Hippel & Basso, Sever, Vieira Connection to the integrands: cluster structure also appeared in on-shell diagrams [Arkahi-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka] [Paulos, Schwab]