Causal Diamonds, Cluster Polytopes and Scattering Amplitudes

Arkani-Hamed, Nima; He, Song; Salvatori, Giulio; Thomas, Hugh

doi:10.48550/arxiv.1912.12948

Cited by 25 publications

(107 citation statements)

References 0 publications

Supporting

Mentioning

106

Contrasting

Order By: Relevance

“…Some examples are shown in Let us point out that in the special case of a cubic theory the 1-loop diagrams have N = n by (2). These may be related to the cluster category of D n quivers [14]. However, the m-cluster category of D n quivers are described in terms of m-angulations of a polygon with (mn − m + 1) sides [16].…”

Section: Planar Diagrams Angulations Of Polygons and Meshesmentioning

confidence: 99%

See 1 more Smart Citation

Towards a categorification of scattering amplitudes

Barmeier¹,

Oak²,

Pal³

et al. 2021

Preprint

View full text Add to dashboard Cite

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate t-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with φ m+2 potentials those corresponding to m-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate t-structures for these types is presented.

show abstract

Section: Planar Diagrams Angulations Of Polygons and Meshesmentioning

confidence: 99%

“…In one line of development scalar field theories with higher order interactions were studied [6][7][8][9][10][11]. Another class of generalization is to obtain similar pictures for higher loop Feynman diagrams [12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Towards a categorification of scattering amplitudes

Barmeier¹,

Oak²,

Pal³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…3 The D n case is completely analogous: by setting all but n of c I as positive constants and requiring all X I ≥ 0, they carve out a generalized associahedron of type D n , which gives the convergence domain, and its boundaries are in one-to-one correspondence with the boundaries of the configuration space that we studied in the previous section. In a close connection with the Y -system, the equations can also be regarded as an evolution in a discrete spacetime with extra boundary conditions [17]. We will come back to such ABHY generalized associahedron as we discuss the pushforward map from considering saddle-point equations.…”

Section: Cluster String Integrals and Their Factorizationmentioning

confidence: 99%

“…. , N Γ , where J|I is the so-called compatibility degree from cluster variable J to I [16,17]. Beyond type A, these degrees go beyond 0, 1 and we have J|I > 0 if and only if the two cluster variables are incompatible (only for simply-laced cases, we have I|J = J|I).…”

Section: Introduction and Reviewmentioning

confidence: 99%

Notes on worldsheet-like variables for cluster configuration spaces

He,

Wang,

Zhang

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli space M 0,n to finite-type cluster algebras. We study worldsheet-like variables, which for classical types have also appeared in the study of the symbol alphabet of Feynman integrals. We provide a systematic derivation of these variables from Y -systems, which allows us to express the u-variables in terms of them and write the corresponding cluster string integrals in compact forms. We mainly focus on the D n type and show how to reach the boundaries of the configuration space, and write the saddle-point equations in terms of these variables. Moreover, these variables make it easier to study various topological properties of the space using a finite-field method. We propose conjectures about quasi-polynomial point count, dimensions of cohomology, and the number of saddle points for the D n space up to n = 10, which greatly extend earlier results.

show abstract

“…So far, the known appearances of cluster algebras in scattering amplitudes can be organized into four broad themes: (1) it has been observed [7] that the singularities of (certain) amplitudes are dictated by cluster variables of the Gr(4, n) cluster algebra; (2) cluster structures appear naturally in the positive Grassmannian description of integrands [8] (and amplituhedra); (3) finite-type cluster algebras provide natural examples of "stringy" integrals that generalize [9] the Koba-Nielsen amplitude [10]; and (4) certain cluster polytopes are amplituhedra for the amplitudes of bi-adjoint φ 3 theory [11].…”

Section: Introductionmentioning

confidence: 99%