This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Conformal field theories have been long known to describe the fascinating universal physics of scale invariant critical points. They describe continuous phase transitions in fluids, magnets, and numerous other materials, while at the same time sit at the heart of our modern understanding of quantum field theory. For decades it has been a dream to study these intricate strongly coupled theories nonperturbatively using symmetries and other consistency conditions. This idea, called the conformal bootstrap, saw some successes in two dimensions but it is only in the last ten years that it has been fully realized in three, four, and other dimensions of interest. This renaissance has been possible both due to significant analytical progress in understanding how to set up the bootstrap equations and the development of numerical techniques for finding or constraining their solutions. These developments have led to a number of groundbreaking results, including world record determinations of critical exponents and correlation function coefficients in the Ising and $O(N)$ models in three dimensions. This article will review these exciting developments for newcomers to the bootstrap, giving an introduction to conformal field theories and the theory of conformal blocks, describing numerical techniques for the bootstrap based on convex optimization, and summarizing in detail their applications to fixed points in three and four dimensions with no or minimal supersymmetry.

The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

Perturbative scattering amplitudes in gauge theories have remarkable simplicity and hidden infinite dimensional symmetries that are completely obscured in the conventional formulation of field theory using Feynman diagrams This suggests the existence of a new understanding for scattering amplitudes where locality and unitarity do not play a central role but are derived consequences from a different starting point In this note we provide such an understanding for N=4 SYM scattering amplitudes in the planar limit, which we identify as ``the volume" of a new mathematical object--the Amplituhedron--generalizing the positive Grassmannian Locality and unitarity emerge hand-in-hand from positive geometry

The Amplituhedron

Unconventional, special-purpose machines may aid in accelerating the solution of some of the hardest problems in computing, such as large-scale combinatorial optimizations, by exploiting different operating mechanisms than those of standard digital computers. We present a scalable optical processor with electronic feedback that can be realized at large scale with room-temperature technology. Our prototype machine is able to find exact solutions of, or sample good approximate solutions to, a variety of hard instances of Ising problems with up to 100 spins and 10,000 spin-spin connections.

/pdf/a-fully-programmable-100-spin-coherent-ising-machine-with-inaxlu4ff3.pdf

A fully programmable 100-spin coherent Ising machine with all-to-all connections

The conformal bootstrap was proposed in the 1970s as a strategy for calculating the properties of second-order phase transitions. After spectacular success elucidating two-dimensional systems, little progress was made on systems in higher dimensions until a recent renaissance beginning in 2008. We report on some of the main results and ideas from this renaissance, focusing on new determinations of critical exponents and correlation functions in the three-dimensional Ising and O(N) models.

The conformal bootstrap

In 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 Conf
 n
 (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 Conf
 n
 (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 classical polylogarithms 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.

/pdf/motivic-amplitudes-and-cluster-coordinates-406iq52ikb.pdf

Motivic amplitudes and cluster coordinates

In 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 Conf_n(P^3) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude A_{7,2} and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster X-coordinates on Conf_n(P^3). 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 A_{7,2} being expressible in terms of classical polylogarithms 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 X-coordinates of a single algebra. In this respect it is similar to Abel's 5-term dilogarithm identity.

/pdf/motivic-amplitudes-and-cluster-coordinates-365gpfl4cm.pdf

Motivic Amplitudes and Cluster Coordinates

Motivated by the cluster structure of two-loop scattering amplitudes in $\mathcal{N}=4$ Yang-Mills theory we define cluster polylogarithm functions. We find that all such functions of weight four are made up of a single simple building block associated with the A(2) cluster algebra. Adding the requirement of locality on generalized Stasheff polytopes, we find that these A(2) building blocks arrange themselves to form a unique function associated with the A(3) cluster algebra. This A(3) function manifests all of the cluster algebraic structure of the two-loop n-particle MHV amplitudes for all n, and we use it to provide an explicit representation for the most complicated part of the n = 7 amplitude as an example.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

https://iopscience.iop.org/article/10.1088/1751-8113/47/47/474005/pdf

Cluster polylogarithms for scattering amplitudes

We apply a bootstrap procedure to two-loop MHV amplitudes in planar $$ \mathcal{N}=4 $$
 super-Yang-Mills theory We argue that the mathematically most complicated part (the Λ2
 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

/pdf/a-cluster-bootstrap-for-two-loop-mhv-amplitudes-1ru6g8j7vk.pdf

A Cluster Bootstrap for Two-Loop MHV Amplitudes

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 = 4 super-Yang-Mills theory. The non-classical part of an amplitude is built from A3 cluster polylogarithm functions; classical polylogarithms with (negative) cluster X - coordinate arguments are added to complete the symbol of the amplitude; beyond-the- symbol terms proportional to � 2 are determined by comparison with the dierential of the amplitude; and the overall additive constant isxed by the collinear limit. We present an explicit formula for the seven-point amplitude R (2) as a sample application.

/pdf/an-analytic-result-for-the-two-loop-seven-point-mhv-1yx8pc0jxt.pdf

John Golden

Papers

Motivic amplitudes and cluster coordinates

Motivic Amplitudes and Cluster Coordinates

Cluster polylogarithms for scattering amplitudes

A Cluster Bootstrap for Two-Loop MHV Amplitudes

An analytic result for the two-loop seven-point MHV amplitude in N = 4 SYM