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Cluster algebras and the subalgebra constructibility of the seven-particle remainder function
https://repo.scoap3.org/record/30173
We review various aspects of cluster algebras and the ways in which they appear in the study of loop-level amplitudes in planar N = 4 $$ \mathcal{N}=4 $$ supersymmetric Yang-Mills theory. In particular, we highlight the different forms of cluster-algebraic structure that appear in this theory’s two-loop MHV amplitudes — considered as functions, symbols, and at the level of their Lie cobracket — and recount how the ‘nonclassical’ part of these amplitudes can be decomposed into specific functions evaluated on the A 2 or A 3 subalgebras of Gr(4 , n ). We then extend this line of inquiry by searching for other subalgebras over which these amplitudes can be decomposed. We focus on the case of seven-particle kinematics, where we show that the nonclassical part of the two-loop MHV amplitude is also constructible out of functions evaluated on the D 5 and A 5 subalgebras of Gr(4 , 7), and that these decompositions are themselves decomposable in terms of the same A 4 function. These nested decompositions take an especially canonical form, which is dictated in each case by constraints arising from the automorphism group of the parent algebra.Golden, JohnFri, 04 Jan 2019 21:27:13 GMThttps://repo.scoap3.org/record/30173urn:ISSN:1029-8479Springer/SISSA2019-01-03Mellin bootstrap for scalars in generic dimension
https://repo.scoap3.org/record/26205
We use the recently developed framework of the Mellin bootstrap to study perturbatively free scalar CFTs in arbitrary dimensions. This approach uses the crossing-symmetric Mellin space formulation of correlation functions to generate algebraic bootstrap equations by demanding that only physical operators contribute to the OPE. We find that there are no perturbatively interacting CFTs with only fundamental scalars in d > 6 dimensions (to at least second order in the perturbation). Our results can be seen as a modest step towards understanding the space of interacting CFTs in d > 6 and are consistent with the intuition that no such CFTs exist.Golden, JohnThu, 14 Jun 2018 17:29:18 GMThttps://repo.scoap3.org/record/26205urn:ISSN:1029-8479Springer/SISSA2018-06-13No unitary bootstrap for the fractal Ising model
https://repo.scoap3.org/record/9783
We consider the conformal bootstrap for spacetime dimension 1 < d < 2. We determine bounds on operator dimensions and compare our results with various theoretical and numerical models, in particular with resummed ϵ -expansion and Monte Carlo simulations of the Ising model on fractal lattices. The bounds clearly rule out that these models correspond to unitary conformal field theories. We also clarify the d → 1 limit of the conformal bootstrap, showing that bounds can be — and indeed are — discontinuous in this limit. This discontinuity implies that for small ϵ = d − 1 the expected critical exponents for the Ising model are disallowed, and in particular those of the d − 1 expansion. Altogether these results strongly suggest that the Ising model universality class cannot be described by a unitary CFT below d = 2. We argue this also from a bootstrap perspective, by showing that the 2 ≤ d < 4 Ising “kink” splits into two features which grow apart below d = 2.Golden, JohnWed, 01 Apr 2015 06:31:55 GMThttps://repo.scoap3.org/record/9783urn:ISSN:1029-8479Springer/SISSA2015-03-31A cluster bootstrap for two-loop MHV amplitudes
https://repo.scoap3.org/record/9025
We apply a bootstrap procedure to two-loop MHV amplitudes in planar N = 4 $$ \mathcal{N}=4 $$ super-Yang-Mills theory. We argue that the mathematically most complicated part (the Λ 2 B 2 coproduct component) of the n -particle amplitude is uniquely determined by a simple cluster algebra property together with a few physical constraints (dihedral symmetry, analytic structure, supersymmetry, and well-defined collinear limits). We present a concise, closed-form expression which manifests these properties for all n .Golden, JohnTue, 03 Feb 2015 12:39:17 GMThttps://repo.scoap3.org/record/9025urn:ISSN:1029-8479Springer/SISSA2015-02-02An analytic result for the two-loop seven-point MHV amplitude in N $$ \mathcal{N} $$ = 4 SYM
https://repo.scoap3.org/record/3825
We describe a general algorithm which builds on several pieces of data available in the literature to construct explicit analytic formulas for two-loop MHV amplitudes in N $$ \mathcal{N} $$ = 4 super-Yang-Mills theory. The non-classical part of an amplitude is built from A 3 cluster polylogarithm functions; classical polylogarithms with (negative) cluster X $$ \mathcal{X} $$ - coordinate arguments are added to complete the symbol of the amplitude; beyond-the-symbol terms proportional to π 2 are determined by comparison with the differential of the amplitude; and the overall additive constant is fixed by the collinear limit. We present an explicit formula for the seven-point amplitude R 7 (2) as a sample application.John GoldenFri, 29 Aug 2014 09:20:41 GMThttps://repo.scoap3.org/record/3825urn:ISSN:1029-8479Springer/SISSA2014-08-27