21 research outputs found
Conformal algebra: R-matrix and star-triangle relation
The main purpose of this paper is the construction of the R-operator which
acts in the tensor product of two infinite-dimensional representations of the
conformal algebra and solves Yang-Baxter equation. We build the R-operator as a
product of more elementary operators S_1, S_2 and S_3. Operators S_1 and S_3
are identified with intertwining operators of two irreducible representations
of the conformal algebra and the operator S_2 is obtained from the intertwining
operators S_1 and S_3 by a certain duality transformation. There are
star-triangle relations for the basic building blocks S_1, S_2 and S_3 which
produce all other relations for the general R-operators. In the case of the
conformal algebra of n-dimensional Euclidean space we construct the R-operator
for the scalar (spin part is equal to zero) representations and prove that the
star-triangle relation is a well known star-triangle relation for propagators
of scalar fields. In the special case of the conformal algebra of the
4-dimensional Euclidean space, the R-operator is obtained for more general
class of infinite-dimensional (differential) representations with nontrivial
spin parts. As a result, for the case of the 4-dimensional Euclidean space, we
generalize the scalar star-triangle relation to the most general star-triangle
relation for the propagators of particles with arbitrary spins.Comment: Added references and corrected typo
New differential equations for on-shell loop integrals
We present a novel type of differential equations for on-shell loop
integrals. The equations are second-order and importantly, they reduce the loop
level by one, so that they can be solved iteratively in the loop order. We
present several infinite series of integrals satisfying such iterative
differential equations. The differential operators we use are best written
using momentum twistor space. The use of the latter was advocated in recent
papers discussing loop integrals in N=4 super Yang-Mills. One of our
motivations is to provide a tool for deriving analytical results for scattering
amplitudes in this theory. We show that the integrals needed for planar MHV
amplitudes up to two loops can be thought of as deriving from a single master
topology. The master integral satisfies our differential equations, and so do
most of the reduced integrals. A consequence of the differential equations is
that the integrals we discuss are not arbitrarily complicated transcendental
functions. For two specific two-loop integrals we give the full analytic
solution. The simplicity of the integrals appearing in the scattering
amplitudes in planar N=4 super Yang-Mills is strongly suggestive of a relation
to the conjectured underlying integrability of the theory. We expect these
differential equations to be relevant for all planar MHV and non-MHV
amplitudes. We also discuss possible extensions of our method to more general
classes of integrals.Comment: 39 pages, 8 figures; v2: typos corrected, definition of harmonic
polylogarithms adde
Generalised ladders and single-valued polylogarithms
We introduce and solve an infinite class of loop integrals which generalises
the well-known ladder series. The integrals are described in terms of
single-valued polylogarithmic functions which satisfy certain differential
equations. The combination of the differential equations and single-valued
behaviour allow us to explicitly construct the polylogarithms recursively. For
this class of integrals the symbol may be read off from the integrand in a
particularly simple way. We give an explicit formula for the simplest
generalisation of the ladder series. We also relate the generalised ladder
integrals to a class of vacuum diagrams which includes both the wheels and the
zigzags.Comment: 27 pages, 7 figure
Systematics of the cusp anomalous dimension
We study the velocity-dependent cusp anomalous dimension in supersymmetric
Yang-Mills theory. In a paper by Correa, Maldacena, Sever, and one of the
present authors, a scaling limit was identified in which the ladder diagrams
are dominant and are mapped onto a Schrodinger problem. We show how to solve
the latter in perturbation theory and provide an algorithm to compute the
solution at any loop order. The answer is written in terms of harmonic
polylogarithms. Moreover, we give evidence for two curious properties of the
result. Firstly, we observe that the result can be written using a subset of
harmonic polylogarithms only, at least up to six loops. Secondly, we show that
in a light-like limit, only single zeta values appear in the asymptotic
expansion, again up to six loops. We then extend the analysis of the scaling
limit to systematically include subleading terms. This leads to a
Schrodinger-type equation, but with an inhomogeneous term. We show how its
solution can be computed in perturbation theory, in a way similar to the
leading order case. Finally, we analyze the strong coupling limit of these
subleading contributions and compare them to the string theory answer. We find
agreement between the two calculations.Comment: 33 pages, 4 figures. Complete LO six-loop result added. Typos
corrected. Version accepted for publicatio
Pan-cancer analysis of whole genomes
Cancer is driven by genetic change, and the advent of massively parallel sequencing has enabled systematic documentation of this variation at the whole-genome scale(1-3). Here we report the integrative analysis of 2,658 whole-cancer genomes and their matching normal tissues across 38 tumour types from the Pan-Cancer Analysis of Whole Genomes (PCAWG) Consortium of the International Cancer Genome Consortium (ICGC) and The Cancer Genome Atlas (TCGA). We describe the generation of the PCAWG resource, facilitated by international data sharing using compute clouds. On average, cancer genomes contained 4-5 driver mutations when combining coding and non-coding genomic elements; however, in around 5% of cases no drivers were identified, suggesting that cancer driver discovery is not yet complete. Chromothripsis, in which many clustered structural variants arise in a single catastrophic event, is frequently an early event in tumour evolution; in acral melanoma, for example, these events precede most somatic point mutations and affect several cancer-associated genes simultaneously. Cancers with abnormal telomere maintenance often originate from tissues with low replicative activity and show several mechanisms of preventing telomere attrition to critical levels. Common and rare germline variants affect patterns of somatic mutation, including point mutations, structural variants and somatic retrotransposition. A collection of papers from the PCAWG Consortium describes non-coding mutations that drive cancer beyond those in the TERT promoter(4); identifies new signatures of mutational processes that cause base substitutions, small insertions and deletions and structural variation(5,6); analyses timings and patterns of tumour evolution(7); describes the diverse transcriptional consequences of somatic mutation on splicing, expression levels, fusion genes and promoter activity(8,9); and evaluates a range of more-specialized features of cancer genomes(8,10-18).Peer reviewe