Motivic amplitudes and cluster coordinates
By J. Golden, A. Goncharov, M. Spradlin, C. Vergu, A. Volovich. 2013
257 citations recorded by the source corpus.
A bstractIn this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Confn(ℙ3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude $ \mathcal{A}_{7,2}^{\mathcal{M}} $ and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster$ \mathcal{X} $-coordinates on Confn(ℙ3). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to $ \mathcal{A}_{7,2}^{\mathcal{M}} $ being expressible in terms of classicalpolylogarithms is most naturally represented by certain quadrilateral faces of the appropriate associahedron. We also find and prove the first known functional equation for the trilogarithm in which all 40 arguments are cluster $ \mathcal{X} $-coordinates of a single algebra. In this respect it is similar to Abel’s 5-term dilogarithm identity.
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/a1858e14ea29e7f9f6230279145c014a5b1b9d08
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Citations in the corpus
- Heptagons from the Steinmann cluster bootstrap (2016, unknown): “[38, 39], the bootstrap approach developed in refs.”
- A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “There is a vast mathematical literature on cluster algebras; we refer the reader to [66] for an introduction focused on amplitudes.”
- Space-time S-matrix and flux tube S-matrix II. Extracting and matching data (2013, background): “It would be interesting to see if this can be a reflection of the recent motivic classification in [55] at the level of the collinear limit.”
- PolyLogTools — polylogs for the masses (2019, background): “J H E P 0 8 ( 2 0 1 9 ) 1 3 5 We see that indeed all the first entries are proportional to π2, and we can immediately read of the values of the coefficients of the terms in the ansatz proportional to π2: In[13]:= sol2 = {c[17] -> 0, c[18] -> 1/12, c[19] -> 0, c[20] -> 0};”
- Cluster algebras (2014, unknown): “Starting in 2013, various authors connected scattering amplitudes of planar N = 4 super Yang-Mills theory to cluster algebras [27, 17, 40, 48].”
- Lectures on differential equations for Feynman integrals (2014, background): “It is interesting to note that some alphabets have been observed to be related to cluster algebras [60–62].”
- The four-loop six-gluon NMHV ratio function (2015, background): “A similar heptagon function bootstrap, based on cluster variables [46, 47] has yielded the (symbol of the) seven-point remainder function — with remarkably little input from physical limits [48].”